On the boundary Carrollian conformal algebra

TL;DR

This paper explores the mathematical structure of the boundary Carrollian conformal algebra (BCCA), constructing modules, analyzing irreducibility, and developing new representations like Whittaker modules to deepen understanding of this infinite-dimensional Lie algebra.

Contribution

It introduces the first detailed study of BCCA modules, irreducibility criteria, and constructs new Whittaker modules, advancing the mathematical understanding of this algebra.

Findings

Constructed modules for BCCA and subalgebras from known modules.

Established irreducibility criteria for BMS$_3$ and BCCA modules.

Developed new Whittaker modules and proved their irreducibility.

Abstract

We initiate the mathematical study of the boundary Carrollian conformal algebra (BCCA), an infinite-dimensional Lie algebra recently discovered in the context of Carrollian physics. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is a filtered but not graded Lie algebra. In this paper, we first construct some modules for the BCCA and one of its subalgebras, which we call $\mathcal{O}$, by restriction of well-known modules of the BMS$_3$ and Witt algebras respectively. Along the way, we prove the irreducibility criteria for the so-called ``induced modules'' of the BMS$_3$ algebra (which we prefer to call massive modules to avoid ambiguity) and show that this is the same criteria for the irreducibility of the Verma modules of the BMS$_3$ algebra. Interestingly, the modules generated by the action of the BCCA on the generating vector of the massive modules are also irreducible under the same criteria. When this criteria holds, every massive module decomposes into a direct sum of two BCCA-submodules, each of which we conjecture to be irreducible. Meanwhile, restricting Verma modules to the BCCA and $\mathcal{O}$ leads to free or ``almost free'' modules, which are not particularly interesting from a representation-theoretic viewpoint. This motivates the construction of BCCA modules intrinsically. To do this, we go through some structure theory on the BCCA to define a new basis and a decreasing filtration on the algebra, using which we construct Whittaker modules over the BCCA and the subalgebra $\mathcal{O}$ and prove criteria for their irreducibility.

Equations

$${}_m\begin{array}{c}=(n-m){\mathcal{O}}_{n+m}-(n+m){\mathcal{O}}_{n-m},\\ =(n-m)P_{n+m}+(n+m)P_{n-m}+\frac{1}{6}n(n^2-1){\delta }_{m,n}{C}_M,\\ =0,\end{array}$$

$$V(h_L,h_M,c_L,c_M)\cong \frac{U(\hat{\mathfrak{b}})}{U(\hat{\mathfrak{b}})\cdot (P_0-2h_M,C_M-c_M)}.$$

$$\tilde{V}(\mathbf{M},\mathbf{s},c_L,c_M)\cong \hat{V}\oplus \hat{W},$$

$$[{\mathcal{O}}_n^{(\lambda )},{\mathcal{O}}_m^{(\lambda )}]=(n-m){\mathcal{O}}_{n+m}^{(\lambda )}-{\lambda }^m(n+m){\mathcal{O}}_{n-m}^{(\lambda )}$$

$$[u_n^{(\lambda )},u_m^{(\lambda )}]=(n-m)(u_{n+m}^{(\lambda )}-4\lambda u_{n+m-2}^{(\lambda )}).$$

$$\mathcal{O}(\lambda )\cong \{\begin{array}{ll}\mathcal{O},& \text{if }\lambda >0,\\ \mathcal{O}(-1),& \text{if }\lambda <0,\end{array}$$

$$[u_n,u_m]$$

$$[u_n,v_m]$$

$$[v_n,v_m]$$

$${\mathcal{F}}_n\mathcal{O}:=\mathrm{span}\{{\mathcal{U}}_k\mid k\ge n\},{\mathcal{F}}_n\mathfrak{b}:=\mathrm{span}\{{\mathcal{U}}_k,{\mathcal{V}}_k\mid k\ge n\}.$$

$$[f\partial ,g\partial ]=(f{g}^{\prime }-f^{\prime }g)\partial ,[L_n,L_m]=(n-m)L_{n+m},$$

$$[f\partial ,g\partial ]=(f{g}^{\prime }-f^{\prime }g)\partial +\frac{1}{12}{\mathrm{Res}}_0(f^{\prime }{g}^{\prime \prime })C,[L_n,L_m]=(n-m)L_{n+m}+\frac{1}{12}n(n^2-1){\delta }_{m,-n}C,$$

$$L_n\cdot I_m=-(a+bn+m)I_{n+m}.$$

$$f\partial \cdot (gd{t}^b)=(f{g}^{\prime }+b{f}^{\prime }g)d{t}^b.$$

$$\begin{array}{c}=(n-m)L_{m+n}+\frac{1}{12}n(n^2-1){\delta }_{m,-n}{C}_L,\\ [L_n,M_m]=(n-m)M_{m+n}+\frac{1}{12}n(n^2-1){\delta }_{m,-n}{C}_M,\\ [M_n,M_m]=0.\end{array}$$

$${}_m\begin{array}{c}=(n-m){\mathcal{O}}_{n+m}-(n+m){\mathcal{O}}_{n-m},\\ =(n-m)P_{n+m}+(n+m)P_{n-m}+\frac{1}{6}n(n^2-1){\delta }_{m,n}{C}_M,\\ =0.\end{array}$$

$$[{\mathcal{O}}_n,{\mathcal{O}}_m]=(n-m){\mathcal{O}}_{n+m}-(n+m){\mathcal{O}}_{n-m}.$$

$${\mathcal{O}}_n\cdot P_m^{(b)}=-(bn+m)P_{n+m}^{(b)}-(bn-m)P_{n-m}^{(b)}(n\ge 1,m\ge 0)$$

$${\mathcal{O}}_n\cdot {\tilde{P}}_m^{(b)}=-(bn+m){\tilde{P}}_{n+m}^{(b)}-(bn-m){\tilde{P}}_{n-m}^{(b)}(n,m\ge 1).$$

$$0$$

$$M_{\phi }:-{\mathrm{Ind}}_{\mathfrak{n}}^{\mathfrak{g}}{\mathbb{k}}_{\phi }$$

$$\cdots \supseteq {\mathfrak{g}}_{-1}\supseteq {\mathfrak{g}}_0\supseteq {\mathfrak{g}}_1\supseteq {\mathfrak{g}}_2\supseteq \cdots$$

$$W_{\phi }:-{\mathrm{Ind}}_{\mathfrak{h}}^{\mathfrak{g}}{\mathbb{k}}_{\phi }$$

$${\mathfrak{g}}^{\phi }:=\{x\in \mathfrak{g}\mid \phi ([x,y])=0\text{ for all }y\in \mathfrak{h}\}.$$

$$J_{\xi }:=U(\mathfrak{g})(y-\xi )\cdot 1_{\phi }$$

$$\hat{\mathfrak{b}}:=\mathrm{span}\{{\mathcal{O}}_n:=L_n-L_{-n},P_m:=M_m+M_{-m},C_M\mid n\ge 1,m\ge 0\}$$

$$\mathrm{Vir}={\mathrm{Vir}}_{+}\oplus {\mathrm{Vir}}_0\oplus {\mathrm{Vir}}_{-},\text{ where }{\mathrm{Vir}}_{\pm }=\underset{n\in {\mathbb{Z}}_{\pm }}{⨁}\mathbb{k}{L}_n\text{ and }{\mathrm{Vir}}_0=\mathbb{k}{L}_0\oplus \mathbb{k}C.$$

$$L_0|h,c⟩=h|h,c⟩,C|h,c⟩=c|h,c⟩,\text{and }{\mathrm{Vir}}_{+}|h,c⟩=0.$$

$$V(h,c):={\mathrm{Ind}}_{{\mathrm{Vir}}_0\oplus {\mathrm{Vir}}_{+}}^{\mathrm{Vir}}{\mathbb{k}}_{h,c}.$$

$$L_{-n_k}\dots L_{-n_1}|h,c⟩,$$

$$V(h,c)\cong U(\mathrm{Vir})/(U(\mathrm{Vir})\cdot (L_{n>0},L_0-h,C-c)).$$

$${\mathcal{O}}_{n_k}\dots {\mathcal{O}}_{n_1}{L}_{m_{\ell }}\dots L_{m_1}{C}^r,$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 The boundary Carrollian conformal algebra

4. 2.2 Whittaker modules

5. 3 Representations of $\mathcal{O}$ and $\widehat{\mathfrak{b}}$ by restriction

6. 3.1 $\mathcal{O}$-modules via restriction of Verma modules

7. 3.2 $\widehat{\mathfrak{b}}$-modules via restriction of Verma modules

8. 3.3 $\widehat{\mathfrak{b}}$-modules via restriction of massive modules

9. 3.4 The $\mathcal{O}$-module $\Omega(\lambda,a)$

10. 4 Alternate basis of the BCCA

11. 4.1 Subalgebras of the Witt algebra

12. 4.2 Change of basis for $\mathcal{O}$

13. 4.3 Change of basis for $\mathcal{P}_{b}$

14. 5 Filtering the BCCA

15. 5.1 Defining the filtration

16. 5.2 The derived subalgebras of the filtered subspaces

17. 5.3 The abelianisations of the filtered subspaces

18. 6 Whittaker modules

19. 6.1 Whittaker modules for $\mathcal{O}$

20. 6.2 Local functions and the orbit method for $\mathcal{O}$

21. 6.3 Whittaker modules for the centreless BCCA

22. 7 Future work

23. A More subalgebras of the Witt algebra

1. Introduction

Over the past two decades, research on infinite-dimensional Lie algebras has enjoyed continuous motivation and inspiration from theoretical physics. Indeed, with the discovery of more examples of infinite-dimensional Lie algebras in physical contexts comes the need to better understand these algebras and their representations. On many occasions, this leads to a productive cycle of progress in both fields, due to the resulting mathematics research informing physics in some manner, which once again motivates another mathematical study. The origin of this paper is no different – we present a first study on another infinite-dimensional Lie algebra that was recently discovered in physics, known as the boundary Carrollian conformal algebra (BCCA), which we denote by $\widehat{\mathfrak{b}}$. Its centreless version is denoted by $\mathfrak{b}$.

The BCCA was discovered as the symmetry algebra of a Carrollian conformal field theory (CCFT) with boundary in [BCC*+*25b], in which it is formulated as a subalgebra of the BMS3 algebra (realised as operators in a quantum field theory) that preserves a choice of boundary on a CCFT on a cylinder. Its original basis consists of the elements $\mathcal{O}_{n}\coloneqq L_{n}-L_{-n}$ ($n\geq 1)$, $P_{n}\coloneqq M_{n}+M_{-n}$ ($n\geq 0$), and $C_{M}$, with its Lie bracket given by

[TABLE]

where $L_{n}$, $M_{n}$, and $C_{M}$ are elements of the BMS3 algebra defined in Definition 2.3.

In fact, the authors of [BCC*+*25b] showed that there exist at least three different boundary-preserving subalgebras of BMS3, but they choose to work with the choice given by (1.1) since it emerges as the algebra of symmetries on the worldsheet of the tensionless open strings, as they demonstrate in their paper. This serves as the primary motivation for our work: understanding the representations of the symmetry algebra of a string theory worldsheet can provide insights into the spectrum of the string theory. Nonetheless, it is interesting to note that one of the other two choices was also discovered in [AFVH25] as the worldsheet symmetry algebra of tropological open strings. This finding presents a further case for the relevance of our work: there may exist connections between the methods developed in this paper and tropical geometry, which in turn could inform string theory.

There are two important features that the BCCA lacks that makes its study both challenging and warranted, these being a grading and a Virasoro subalgebra. Symmetry algebras of string theory worldsheets usually have both, making them amenable to thorough, quantum mechanical treatment using the power tools of vertex operator algebras and semi-infinite cohomology [BPZ84, FGZ86, Bor86, FLM88, LZ93]. Today, these toolkits can be coupled with the troves of knowledge on the representation theory of integer-graded infinite-dimensional Lie algebras, to which the the past two decades of active research in both physics [BGMM10, BJMN16, BJL*+*16, CGOR16, BCG17, FPSSJ19, GPSJ*+*20, BCG20, FOV25, BFOGV24, ABD*+*25] and mathematics [ZD09, LZ10, GJP11, BJP11, Rad13, MZ14, JZ15, LZ16, ALZ16, AR16, JPZ18, BKRS21, LPX22, LPXZ24, DGL24] have contributed. However, with neither a grading nor a Virasoro subalgebra, the BCCA cannot be studied using these well-established techniques. Hence, understanding the representations of the BCCA presents a novel challenge for both physicists and mathematicians. We therefore develop and employ new approaches that we hope can lay the groundwork for future studies on infinite-dimensional Lie algebras that are not integer-graded while shedding some light on the spectrum of the open tensionless string.

In this paper, we start by studying representations of the BCCA and its subalgebra $\mathcal{O}\coloneqq\operatorname{span}\{\mathcal{O}_{n}\mid n\geq 1\}$ by restricting well-known representations of the Virasoro and BMS3 algebras, namely the tensor density modules, Verma modules, the so-called “induced modules” introduced in [BO14] which we prefer to call “massive modules” (see [CGOR16]), and the modules $\Omega(\lambda,a)$ introduced in [LZ14]. We work over an arbitrary, characteristic-zero field $\Bbbk$ to prove the following.

Theorem 1.1** (Lemma 2.7, Propositions 3.3, 3.5, 3.12, and 3.19).**

Let $h_{L},h_{M},c_{L},c_{M},a,b,\mathbf{M},\mathbf{s}\in\Bbbk$, and $\lambda\in\Bbbk^{*}$. Then the following hold.

1. (1)

The tensor density module $I(0,b)$ (see Definition 2.2) decomposes into a direct sum of two $\mathcal{O}$-submodules. 2. (2)

When restricted to $\mathcal{O}$, the Virasoro Verma module $V(h_{L},c_{L})$ (see Definition 3.1) is a free module of rank 1. That is, $V(h_{L},c_{L})\cong U(\mathcal{O})$ as $\mathcal{O}$-modules. 3. (3)

When restricted to the BCCA, the BMS3* Verma module $V(h_{L},h_{M},c_{L},c_{M})$ (see Definition 3.4) has the following form:*

[TABLE] 4. (4)

The BCCA-module $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}$ (see Definition 3.7) is irreducible if and only if $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n$. 5. (5)

The $\mathcal{O}$-module $\Omega(\lambda,a)$ (see Subsection 3.4) is irreducible if and only if $\lambda\neq\pm 1$ and $a\neq 0$.

Taking Theorem 1.1(4) a bit further, we showed that the massive module $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ is decomposable as a $\widehat{\mathfrak{b}}$-modules when $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}$ is irreducible (i.e., $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n$).

Theorem 1.2** (Propositions 3.14 and 3.16, Corollary 3.17).**

Let $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n$. Then the massive module $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ decomposes into a direct sum of two $\widehat{\mathfrak{b}}$-submodules:

[TABLE]

where $\widehat{V}$ and $\widehat{W}$ are given by Notation 3.13 and Definition 3.15.

We then present the following conjecture as an avenue for further study.

Conjecture 1.3** (Conjecture 3.18).**

The $\widehat{\mathfrak{b}}$-modules $\widehat{V}$ and $\widehat{W}$ are indecomposable.

On the way to proving Theorem 1.1(4), we also characterise when the massive modules over the BMS3 algebra111These are known as induced modules in the physics literature, with the name “massive modules” reserved for a subset of them. In this paper, we choose to refer to all modules given in Definition 3.7 as massive modules. are irreducible, which to our knowledge was not known before.

Theorem 1.4** (Theorem 3.8).**

Let $\mathbf{M},\mathbf{s},c_{L},c_{M}\in\Bbbk$. Then the massive module $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ over the BMS3 algebra (see Definition 3.7) is irreducible if and only if $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n$.

This is in line with what is expected based on the gravitational interpretation of the coadjoint orbits of the BMS3 group (see [BO15] for more details).

Theorem 1.1 shows that many of the well-known modules over the Virasoro and BMS3 algebras become free, or “almost free” when restricted to the BCCA. From the perspective of representation theory, these are not the most interesting BCCA-modules, so we later shift our focus to studying the representation theory of the BCCA intrinsically, by way of Whittaker modules (see Definition 2.9). The main issue is that the presentation of the BCCA given in (1.1) makes it difficult to even define what a Whittaker module over the BCCA should be. For this reason, we spend some time constructing a different basis of the (centreless) BCCA which better lends itself to the construction of Whittaker modules.

The change of basis is achieved by analysing the subalgebra $\mathcal{O}$, which is a subalgebra of the Witt algebra $\mathcal{W}\coloneqq\operatorname{Der}(\Bbbk[t,t^{-1}])$, using techniques from [Buz23, BB25]. In fact, we work in a more general setting: for $\lambda\in\Bbbk^{*}$ and $n\in\mathbb{Z}$, let $\mathcal{O}^{(\lambda)}_{n}\coloneqq L_{n}-\lambda^{n}L_{-n}$, and define $\mathcal{O}(\lambda)\coloneqq\operatorname{span}\{\mathcal{O}^{(\lambda)}_{n}\mid n\geq 1\}$. It is easy to see that $\mathcal{O}(\lambda)$ is a Lie algebra with bracket given by

[TABLE]

for all $n,m\in\mathbb{Z}_{+}$ (see Lemma 4.14).

Theorem 1.5** (Theorem 4.16, Proposition 4.22, and Corollary 4.23).**

Let $\lambda,\mu\in\Bbbk^{*}$. Then $\mathcal{O}(\lambda)$ has another basis $\{u^{(\lambda)}_{n}\mid n\geq 1\}$ with Lie bracket

[TABLE]

Consequently, $\mathcal{O}(\lambda)\cong\mathcal{O}(\mu)$ if and only if $\sqrt{\frac{\lambda}{\mu}}\in\Bbbk$. In particular, if $\Bbbk$ is algebraically closed (for example, if $\Bbbk=\mathbb{C}$), then $\mathcal{O}(\lambda)\cong\mathcal{O}$ for all $\lambda\in\Bbbk$. If $\Bbbk=\mathbb{R}$, then

[TABLE]

but $\mathcal{O}(-1)\not\cong\mathcal{O}$.

We further present some consequences of Theorem 1.5 in Section 4. For example, it becomes easy to show that $\mathcal{O}$ is not a simple Lie algebra (in fact, it is not even perfect) – see Corollary 4.19. Certainly, the same is true for both $\mathfrak{b}$ and $\widehat{\mathfrak{b}}$.

Theorem 1.5 tells us that the Lie subalgebras $\mathcal{O}$ and $\mathcal{O}(-1)$ of the Witt algebra are isomorphic as complex Lie algebras but not as real ones. This leads to an interesting physical implication. In [BCC*+*25b], the authors start with the BMS3 algebra realised as vector fields which generate the infinitesimal transformations that leave the Carrollian structure on a cylinder (given by a symmetric $(0,2)$-tensor field $h$ that is corank-1 everywhere on the cylinder and a nowhere-vanishing vector field $\xi$ such that $\ker h$ is generated by $\xi$) invariant up to an overall conformal factor. Using this, the Lie algebra $\mathcal{O}$ was constructed as the subalgebra of the Witt algebra consisting of the vector fields that preserve a choice of boundaries on the cylinder at [math] and $\pi$. Likewise, the Lie algebra $\mathcal{O}(-1)$ can be viewed as the analogous Lie algebra when the boundaries are instead placed at $\tfrac{\pi}{2}$ and $\tfrac{3\pi}{2}$. But one can always recover the original boundary conditions by translating the periodic coordinate by $\tfrac{\pi}{2}$, which puts back in the setting where $\mathcal{O}$ is the distinguished boundary-preserving subalgebra. This equivalence must be reflected algebraically too, which indicates that $\mathcal{O}(-1)$ and $\mathcal{O}$ must be isomorphic Lie algebras. Thus, by Theorem 1.5, we confirm that the authors [BCC*+*25b] must be working over $\mathbb{C}$ and not $\mathbb{R}$. We provide details of this observation in Remark 4.24.

Having constructed the alternate basis of $\mathcal{O}$ in Theorem 1.5, we then extend it to a new basis of the full centreless BCCA.

Theorem 1.6** (Corollary 4.34).**

There is another basis of $\mathfrak{b}$ denoted $\{u_{n},v_{m}\mid n\geq 1,m\geq 0\}$. The bracket of the centreless BCCA with this basis is given by

[TABLE]

Using the new basis for $\mathfrak{b}$ from Theorem 1.6, we can easily construct a filtration for the centreless BCCA, allowing us to define and study Whittaker modules for $\mathfrak{b}$, since filtered Lie algebras always have Whittaker modules (see Lemma 2.11). For convenience, we shift the basis as follows: let $\mathcal{U}_{n}\coloneqq u_{n+2}$ and $\mathcal{V}_{n}\coloneqq v_{n+1}$ for $n\geq-1$.

Proposition 1.7** (Proposition 5.2).**

For $n\geq-1$, define

[TABLE]

Then $\mathcal{F}$ is a filtration of $\mathcal{O}$ and $\mathfrak{b}$.

Consequently, we define Whittaker functions on $\mathfrak{b}$ to be Lie algebra homomorphisms $\psi_{n}\colon\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$ for some $n\in\mathbb{N}$. Given a Whittaker function $\psi_{n}\colon\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$, we let $\Bbbk_{\psi_{n}}=\Bbbk 1_{\psi_{n}}$ be the one-dimensional $\mathcal{F}_{n}\mathfrak{b}$-module defined by $x\cdot 1_{\psi_{n}}=\psi_{n}(x)1_{\psi_{n}}$ for $x\in\mathcal{F}_{n}\mathfrak{b}$. The universal Whittaker module of type $\psi_{n}$ is defined to be $M_{\psi_{n}}\coloneqq\operatorname{Ind}_{\mathcal{F}_{n}\mathfrak{b}}^{\mathfrak{b}}\Bbbk_{\psi_{n}}$. Similar definitions can be made for the subalgebra $\mathcal{O}$.

Theorem 1.8** (Theorems 6.6 and 6.28).**

Let $n\in\mathbb{N}$, and let $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ and $\psi_{n}\colon\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$ be Whittaker functions. Then the following hold.

1. (1)

If $n=0$, then the $\mathcal{O}$-module $M_{\varphi_{0}}$ is irreducible if and only if $\varphi_{0}(\mathcal{U}_{2})\neq 4\varphi_{0}(\mathcal{U}_{0})$. 2. (2)

If $n\geq 1$, then the $\mathcal{O}$-module $M_{\varphi_{n}}$ is irreducible if and only if $\varphi_{n}(\mathcal{U}_{2n+1})\neq 4\varphi_{n}(\mathcal{U}_{2n-1})$ or $\varphi_{n}(\mathcal{U}_{2n+2})\neq 4\varphi_{n}(\mathcal{U}_{2n})$. 3. (3)

If $n=0$, then the $\mathfrak{b}$-module $M_{\psi_{0}}$ is irreducible if and only if $\psi_{0}(\mathcal{V}_{0})\neq 0$. 4. (4)

If $n\geq 1$, then the $\mathfrak{b}$-module $M_{\psi_{n}}$ is irreducible if and only if $\psi_{n}(\mathcal{V}_{2n})\neq 0$ or $\psi_{n}(\mathcal{V}_{2n-1})\neq 0$.

Certainly, there exists strong physical and mathematical motivation for this paper. The former arises from the fact that an infinite-dimensional Lie algebra which is not integer-graded has made an appearance in a modern theoretical physics setting, meaning that we need to develop the tools needed to study such objects thoroughly, starting with the BCCA as the inaugural case study. The latter comes from the desire to understand infinite-dimensional subalgebras of infinite-dimensional Lie algebras more thoroughly.

This paper is organised as follows. In Section 2, we set up notation, present key definitions along with minor or known results, and introduce the main objects of study of this paper. Section 3 constructs modules of the BCCA and one of its subalgebras by restricting well-known modules of the BMS3 and Virasoro algebras. This would be the naive approach to the studying the representations of the BCCA based on its physical origins in [BCC*+*25b]. The emergence of free and “almost free” modules motivates the need to study modules of the BCCA intrinsically. This requires the change of basis presented in Section 4, from which we can construct a decreasing filtration of the BCCA and compute derived subalgebras, as done in Section 5. All this effort pays off in Section 6, in which we construct the universal Whittaker modules over the BCCA and its subalgebra $\mathcal{O}$, and determine criteria for their irreducibility. Finally, in Section 7, we present a myriad of future directions of work that can be undertaken on the representation theory of the BCCA, from both physical and mathematical perspectives.

Acknowledgements: The authors would like to thank Blagoje Oblak for fascinating, insightful discussions on the induced representations of the BMS3 group and algebra, Martin Schlichenmaier for interesting discussions about the relationship between the BCCA and Krichever–Novikov algebras, and José Figueroa-O’Farrill for comments on a draft of this paper. GSV would also like to thank Daniel Grumiller and Priyadarshini Pandit for discussions on the BCCA and the tensionless open string during the conference titled “Non-Lorentzian Geometries and their Applications”, which took place at the Hamilton Mathematics Institute, Trinity College Dublin, from 29th April to 2nd May 2025. GSV is supported by the Science and Technologies Facilities Council studentship [grant number 2615874]. XH is supported by NSFC (No. 12201027). HT is partially supported by NSFC (No.12271085) and CSC (No.202506620188). KZ is partially supported by NSERC (311907-2020). TAP is a PhD student at the University of Edinburgh with Susan J. Sierra.

2. Preliminaries

Throughout this paper, we use the following notation:

• •

$\Bbbk$ denotes an arbitrary field of characteristic zero.

• •

$\Bbbk^{*}$ denotes the set of nonzero elements of the field $\Bbbk$.

• •

$\mathbb{N}$ denotes the set of non-negative integers (including zero).

• •

$\mathbb{Z}_{+}$ and $\mathbb{Z}_{-}$ denote the set of positive and negative integers, respectively.

All Lie algebras and vector spaces are considered over $\Bbbk$ unless otherwise specified. We will also abbreviate $\otimes_{\Bbbk}$ to $\otimes$.

2.1. The boundary Carrollian conformal algebra

We define the fundamental objects of interest in this paper and present the construction of the boundary Carrollian conformal algebra (BCCA).

Definition 2.1**.**

The Witt algebra is defined as $\mathcal{W}\coloneqq\operatorname{Der}(\Bbbk[t,t^{-1}])=\Bbbk[t,t^{-1}]\partial$, where $\partial\coloneqq\frac{d}{dt}$. The standard basis of the Witt algebra is $L_{n}\coloneqq-t^{n+1}\partial$ ($n\in\mathbb{Z})$, and its Lie bracket is given by

[TABLE]

for all $f,g\in\Bbbk[t,t^{-1}]$ and $n,m\in\mathbb{Z}$. The Witt algebra $\mathcal{W}$ can be seen as the Lie algebra of algebraic vector fields on $\Bbbk^{*}$.

The Virasoro algebra $\operatorname{Vir}$ is the universal central extension of the Witt algebra. As a vector space, $\operatorname{Vir}=\mathcal{W}\oplus\Bbbk C$ with bracket given by

[TABLE]

for all $f,g\in\Bbbk[t,t^{-1}]$, where $C$ is central, and $\operatorname{Res}_{0}(h)$ is the residue of $h\in\Bbbk[t,t^{-1}]$ at zero, or in other words, the coefficient of $t^{-1}$ in $h$.

The one-sided Witt algebra is defined as $\mathbb{W}_{1}\coloneqq\operatorname{Der}(\Bbbk[t])=\Bbbk[t]\partial$. Equivalently, it can be viewed as the subalgebra of $\mathcal{W}$ spanned by $\{L_{n}\mid n\geq-1\}$ and it is the Lie algebra of algebraic vector fields on $\Bbbk$.

We often switch between the basis-free notation $f\partial$ for elements of $\mathcal{W}$ and the basis notation $L_{n}$.

Definition 2.2**.**

For $a,b\in\Bbbk$, define a $\mathcal{W}$-module $I(a,b)=\operatorname{span}\{I_{n}\mid n\in\mathbb{Z}\}$ as follows:

[TABLE]

The modules $I(a,b)$ are known as tensor density modules. In basis-free notation, we define $I(a,b)\coloneqq t^{a-b}\Bbbk[t,t^{-1}]\ dt^{b}$ with $\mathcal{W}$-action given by

[TABLE]

With this perspective, we have $I_{n}\coloneqq-t^{n+a-b}\ dt^{b}$. Note that $I(0,-1)$ is the adjoint module of $\mathcal{W}$.

We may then construct the two-parameter family of Lie algebras $\mathcal{W}(a,b)\coloneqq\mathcal{W}\ltimes I(a,b)$. The special case $\mathcal{W}(0,-1)$ is the centreless version of the BMS3 algebra, presented in the next definition.

Definition 2.3**.**

The BMS3 algebra, denoted $\mathfrak{bms}$, is the universal central extension of $\mathcal{W}(0,-1)$. It is the infinite dimensional Lie algebra spanned by $\{L_{n},M_{n}\mid n\in\mathbb{Z}\}$ and two central elements $C_{L},C_{M}$, with Lie bracket

[TABLE]

Here, $\{L_{n}\mid n\in\mathbb{Z}\}$ and $C_{L}$ form a basis of a Virasoro subalgebra and $\{M_{n}\mid n\in\mathbb{Z}\}$ is the basis of $I(0,-1)$ presented in Definition 2.2. The BMS3 algebra was introduced in mathematics as the $W(2,2)$ algebra222Not to be confused with our $\mathcal{W}(a,b)$ algebras. by Zhang and Dong but with $C_{L}=C_{M}$ [ZD09].

We now define the BCCA in its original form given by [BCC*+*25b, Equation (7)]. This will be the main object of study of our paper.

Definition 2.4**.**

Let $\mathcal{O}_{n}\coloneqq L_{n}-L_{-n}$ and $P_{n}\coloneqq M_{n}+M_{-n}$ for all $n\in\mathbb{Z}$, which means that $\mathcal{O}_{n}=-\mathcal{O}_{-n}$ and $\mathcal{P}_{n}=\mathcal{P}_{-n}$ for all $n\geq 1$. The boundary Carrollian conformal algebra (BCCA), denoted $\widehat{\mathfrak{b}}$, is the Lie subalgebra of $\mathfrak{bms}$ with basis $\{\mathcal{O}_{n},P_{m},C_{M}\mid n\geq 1,\ m\geq 0\}$ and Lie bracket

[TABLE]

In this paper, we will mostly work with the centreless BCCA, which we call $\mathfrak{b}$, and reserve $\widehat{\mathfrak{b}}$ to refer to the centrally extended version of $\mathfrak{b}$. In other words, we set $C_{M}=0$ in $\mathfrak{b}$.

Remark 2.5**.**

The Lie algebra $\mathfrak{b}$ is the fixed subalgebra of $\mathcal{W}(0,-1)$ under the automorphism $L_{n}\mapsto-L_{-n},\,M_{n}\mapsto M_{-n}$.

First, we seek to write $\mathfrak{b}$ as a semi-direct sum in a similar manner to how the centreless BMS3 algebra can be written as the semi-direct sum $\mathcal{W}(0,-1)\coloneq\mathcal{W}\ltimes I(0,-1)$. To do this, we define the following subspaces of $\mathcal{W}$ and $I(a,b)$.

Definition 2.6**.**

The subalgebra $\mathcal{O}\subseteq\mathcal{W}$ is spanned by $\{\mathcal{O}_{n}\mid n\geq 1\}$, where $\mathcal{O}_{n}\coloneqq L_{n}-L_{-n}$. Its Lie bracket is given by

[TABLE]

Let $I(0,b)=\operatorname{span}\{I_{n}\mid n\in\mathbb{Z}\}$ be the tensor density modules from Definition 2.2. Let $P^{(b)}_{n}\coloneqq I_{n}+I_{-n}$ and construct $\mathcal{P}_{b}\coloneqq\operatorname{span}\{P^{(b)}_{n}\mid n\in\mathbb{N}\}$. Likewise, let $\widetilde{P}^{(b)}_{n}\coloneqq I_{n}-I_{-n}$ and construct $\widetilde{\mathcal{P}}_{b}\coloneqq\operatorname{span}\{\widetilde{P}^{(b)}_{n}\mid n\in\mathbb{N}^{+}\}$. The vector spaces $\mathcal{P}_{b}$ and $\widetilde{\mathcal{P}}_{b}$ are $\mathcal{O}$-modules under the following actions of $\mathcal{O}$ inherited from the action of $\mathcal{W}$ on $I(0,b)$:

[TABLE]

For $b=-1$, we write $\mathcal{P}\coloneqq\mathcal{P}_{-1}$. Note that $\widetilde{\mathcal{P}}_{-1}$ is the adjoint module of $\mathcal{O}$.

It is easy to see that $I(0,b)$ decomposes as a direct sum of $\mathcal{P}_{b}$ and $\widetilde{\mathcal{P}}_{b}$ as an $\mathcal{O}$-module.

Lemma 2.7**.**

As $\mathcal{O}$-modules, $I(0,b)\cong\mathcal{P}_{b}\oplus\widetilde{\mathcal{P}}_{b}$.

Proof.

There exists an $\mathcal{O}$-module homomorphism $\rho\colon I(0,b)\to\mathcal{P}_{b}$ given by $\rho(I_{n})=P_{n}$ for all $n\in\mathbb{Z}$. We then have a short exact sequence of $\mathcal{O}$-modules

[TABLE]

The inclusion $\mathcal{P}_{b}\hookrightarrow I(0,b)$ is a section of the short exact sequence, and thus the sequence is split. It follows that $I(0,b)\cong\mathcal{P}_{b}\oplus\widetilde{\mathcal{P}}_{b}$ for all $b\in\Bbbk$. ∎

We therefore see that Definition 2.4 can be succinctly summarised as $\mathfrak{b}\coloneqq\mathcal{O}\ltimes\mathcal{P}_{-1}=\mathcal{O}\ltimes\mathcal{P}$, analogous to how the centreless BMS3 algebra is defined as $\mathcal{W}(0,-1)\coloneqq\mathcal{W}\ltimes I(0,-1)$.

2.2. Whittaker modules

Whittaker modules were first discovered for $\mathfrak{sl}_{2}$ by Arnal and Pinczon in [AP74], following which Kostant defined them for finite-dimensional complex semisimple Lie algebras in [Kos78]. This was then generalised to arbitrary Lie algebras with triangular decomposition. Today, Whittaker modules are defined more generally via the notion of “Whittaker pairs” [BM11, MZ14] and play an important role in the representation theory of various infinite-dimensional Lie algebras [Chr08, OW09, ZTL10, GL11, Wan11, LWZ10, ALZ16, LPX19, CJ20, Che21, DGL24, CGLW24].

In this paper, we construct Whittaker modules of $\mathcal{O}$ and $\mathfrak{b}$ based on the definition given in [BM11, Section 3.2] and [MZ14, Section 5.1], thereby presenting explicit examples of their constructions for infinite-dimensional Lie algebras that are not integer-graded.

Definition 2.8**.**

Let $\mathfrak{g}$ be a Lie algebra.

1. (1)

Define $\mathfrak{g}^{0}\coloneq\mathfrak{g}$ and $\mathfrak{g}^{k+1}\coloneq[\mathfrak{g}^{k},\mathfrak{g}]$ for all $k\in\mathbb{N}$. We say that $\mathfrak{g}$ is quasi-nilpotent if $\bigcap_{k=0}^{\infty}\mathfrak{g}^{k}=0$. 2. (2)

A $\mathfrak{g}$-module $V$ is said to be locally nilpotent if for every $v\in V$, there exists $s(v)\in\mathbb{N}$ such that $x_{1}x_{2}\dots x_{s(v)}\cdot v=0$ for all $x_{1},\dots,x_{s(v)}\in\mathfrak{g}$.

Equipped with the above definition, we can now define Whittaker modules.

Definition 2.9**.**

Let $\mathfrak{g}$ be a Lie algebra with a quasi-nilpotent subalgebra $\mathfrak{n}\subseteq\mathfrak{g}$ such that $\mathfrak{g}/\mathfrak{n}$ is a locally nilpotent $\mathfrak{n}$-module under the adjoint action, and let $V$ be a $\mathfrak{g}$-module.

1. (1)

The $\mathfrak{g}$-module $V$ is a Whittaker module if $U(\mathfrak{n})\cdot v$ is finite-dimensional for all $v\in V$ (in other words, $V$ is a locally finite $\mathfrak{n}$-module). 2. (2)

A Lie algebra homomorphism $\varphi\colon\mathfrak{n}\to\Bbbk$ is called a Whittaker function (so $\varphi([\mathfrak{n},\mathfrak{n}])=0$). A vector $v\in V$ is called a Whittaker vector of type $\varphi$ if $x\cdot v=\varphi(x)v$ for all $x\in\mathfrak{n}$. The $\mathfrak{g}$-module $V$ is called a Whittaker module of type $\varphi$ if it is generated by a Whittaker vector of type $\varphi$. 3. (3)

Let $\Bbbk_{\varphi}\coloneq\Bbbk 1_{\varphi}$ be the one-dimensional $\mathfrak{n}$-module defined by $\varphi$, in other words, $x\cdot 1_{\varphi}=\varphi(x)1_{\varphi}$ for all $x\in\mathfrak{n}$. Then the induced module

[TABLE]

is the universal Whittaker module of type $\varphi$.

We proceed to briefly show how Definition 2.9 can be readily applied to filtered Lie algebras.

Definition 2.10**.**

A filtered Lie algebra is a Lie algebra $\mathfrak{g}$ with a decreasing sequence of subalgebras of $\mathfrak{g}$

[TABLE]

such that $[\mathfrak{g}_{i},\mathfrak{g}_{j}]\subseteq\mathfrak{g}_{i+j}$. We further demand that the following two conditions are satisfied:

1. (1)

The filtration is weakly convergent. That is, $\bigcap_{k=0}^{\infty}\mathfrak{g}_{k}=0$. 2. (2)

The filtration is bounded from below. That is, there exists $N\in\mathbb{Z}$ such that $\mathfrak{g}_{k}=\mathfrak{g}$ for all $k<N$.

Using this definition, it is easy to show that filtered Lie algebras always have Whittaker modules.

Lemma 2.11**.**

Let $\mathfrak{g}$ be a filtered Lie algebra as per Definition 2.10. If $n\geq 1$, then $\mathfrak{g}_{n}$ is a quasi-nilpotent Lie algebra, and $\mathfrak{g}/\mathfrak{g}_{n}$ is a locally nilpotent $\mathfrak{g}_{n}$-module.

In particular, one can always construct Whittaker modules for $\mathfrak{g}$ by choosing $\mathfrak{n}\coloneqq\mathfrak{g}_{n}$ in Definition 2.9 for any $n\geq 1$.

Proof.

Use conditions (1) and (2) from Definition 2.10. ∎

We introduce a related but slightly different class of modules known as quasi-Whittaker modules that were recently studied in [CGL*+*25].

Definition 2.12**.**

Let $\mathfrak{h}$ be a non-perfect ideal of a non-semisimple Lie algebra $\mathfrak{g}$, let $\varphi\colon\mathfrak{h}\to\Bbbk$ be a Lie algebra homomorphism, and let $V$ be a $\mathfrak{g}$-module.

1. (1)

A vector $v\in V$ is called a quasi-Whittaker vector of type $\varphi$ if $h\cdot v=\varphi(h)v$ for all $h\in\mathfrak{h}$. The $\mathfrak{g}$-module $V$ is called a quasi-Whittaker module of type $\varphi$ if it is generated by a Whittaker vector of type $\varphi$. 2. (2)

Let $\Bbbk_{\varphi}\coloneq\Bbbk 1_{\varphi}$ be the one-dimensional $\mathfrak{h}$-module defined by $\varphi$. Then the induced module

[TABLE]

is the universal quasi-Whittaker module of type $\varphi$.

Remark 2.13**.**

Definition 2.12 applies to $\mathfrak{h}$, which need not be quasi-nilpotent, as opposed to $\mathfrak{n}$ in Definition 2.9. However, it requires that $\mathfrak{h}$ is an ideal of $\mathfrak{g}$, while $\mathfrak{n}$ need not be.

Next, we define the notion of the Whittaker annihilator corresponding to a Whittaker function, a key concept introduced in [CGL*+*25].

Definition 2.14**.**

Let $\mathfrak{h}$ be a non-perfect ideal of a non-semisimple Lie algebra $\mathfrak{g}$, and let $\varphi\colon\mathfrak{h}\to\Bbbk$ be a Lie algebra homomorphism. The Whittaker annihilator of $\varphi$ over $\mathfrak{g}$ is defined to be

[TABLE]

We conclude this section by presenting two key results from [CGL*+*25] which emphasise the importance of the Whittaker annihilator.

Theorem 2.15** ([CGL*+*25, Theorem 3.3 and Corollary 3.12]).**

Let $\mathfrak{g}$ be non-semisimple Lie algebra, $\mathfrak{h}$ a non-perfect ideal of $\mathfrak{g}$, and $\varphi\colon\mathfrak{h}\to\Bbbk$ a Lie algebra homomorphism. Then the following hold.

1. (1)

The universal quasi-Whittaker module $W_{\varphi}=\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}\Bbbk_{\varphi}$ is irreducible if and only if $\mathfrak{g}^{\varphi}=\mathfrak{h}$. 2. (2)

If $\dim(\mathfrak{g}^{\varphi}/\mathfrak{h})=1$, choose $y\in\mathfrak{g}^{\varphi}\setminus\mathfrak{h}$, so that $\mathfrak{g}^{\varphi}=\mathfrak{h}\oplus\Bbbk y$. Then all maximal submodules of $W_{\varphi}$ are of the form

[TABLE]

for $\xi\in\Bbbk$.

3. Representations of $\mathcal{O}$ and $\widehat{\mathfrak{b}}$ by restriction

Recall that in its original form presented in [BCC*+*25b], the BCCA emerged as the subalgebra

[TABLE]

of the BMS3 algebra. Hence, the most natural course of action is to study the restrictions of representations of $\operatorname{Vir}$ and $\mathfrak{bms}$ to $\mathcal{O}$ and $\widehat{\mathfrak{b}}$ respectively. In doing so, we will also motivate the need for the alternative, intrinsic approaches of constructing representations of $\mathfrak{b}$ presented in this paper. Note that the representation theory of $\operatorname{Vir}$ and $\mathfrak{bms}$ is fairly extensive. Hence, we restrict our attention to the most popular choices of modules over these algebras, namely the Verma modules (over both) and the so-called massive modules (over $\mathfrak{bms}$).

3.1. $\mathcal{O}$-modules via restriction of Verma modules

We first recall the definition of a Verma module of the Virasoro algebra. Recalling Definition 2.1, the Virasoro algebra admits a triangular decomposition

[TABLE]

Definition 3.1**.**

Let $h,c\in\Bbbk$. Define the one-dimensional $\operatorname{Vir}_{0}\oplus\operatorname{Vir}_{+}$-module $\Bbbk_{h,c}\coloneqq\Bbbk\ket{h,c}$, where

[TABLE]

The Verma module of the Virasoro algebra $V(h,c)$ is the $\operatorname{Vir}$-module induced from $\Bbbk_{h,c}$. That is,

[TABLE]

Remark 3.2**.**

For all $h,c\in\Bbbk$, the module $V(h,c)$ admits a Poincaré–Birkhoff–Witt (PBW) basis of monomials

[TABLE]

where $n_{k}\geq n_{k-1}\geq\dots\geq n_{1}\geq 1$. Therefore, $V(h,c)$ is cyclically generated by the action of $\operatorname{Vir}_{-}$ on $\ket{h,c}$. As left $U(\operatorname{Vir})$-modules,

[TABLE]

The following proposition describes $V(h,c)$ as an $\mathcal{O}$-module.

Proposition 3.3**.**

The Verma module $V(h,c)$ is a free $U(\mathcal{O})$-module of rank 1 for all $h,c\in\Bbbk$.

Proof.

Consider the basis $\{\mathcal{O}_{n},L_{m},C\mid n\geq 1,m\geq 0\}$ of $\operatorname{Vir}$. This then induces a different PBW basis on $U(\operatorname{Vir})$:

[TABLE]

where $1\leq n_{1}\leq\dots\leq n_{k}$, $0\leq m_{1}\leq\dots\leq m_{\ell}$, and $r\in\mathbb{N}$. In this PBW basis, the description of $V(h,c)$ given in Remark 3.2 as a quotient of $U(\operatorname{Vir})$ by a left ideal readily translates to the fact that

[TABLE]

is a basis for $V(h,c)$ with no relations on the $\mathcal{O}_{n}$s. This proves the proposition. ∎

Free modules are not the most interesting from a representation theoretic point of view as they are not irreducible. This motivates our approach in the later sections, where we change basis and construct $\mathcal{O}$-modules as modules of a subalgebra of $\mathbb{W}_{1}$ that is isomorphic to $\mathcal{O}$. This will give us explicit examples of classes of irreducible $\mathcal{O}$-modules.

3.2. $\widehat{\mathfrak{b}}$-modules via restriction of Verma modules

Just like the Virasoro algebra, the BMS3 algebra is also a graded Lie algebra: we have $\mathfrak{bms}=\bigoplus_{n\in\mathbb{Z}}\mathfrak{bms}_{n}$, where $\mathfrak{bms}_{n}\coloneqq\Bbbk L_{n}\oplus\Bbbk M_{n}$ for $n\neq 0$, and $\mathfrak{bms}_{0}\coloneqq\Bbbk L_{0}\oplus\Bbbk M_{0}\oplus\Bbbk C_{L}\oplus\Bbbk C_{M}$. Its triangular decomposition is given by $\mathfrak{bms}=\mathfrak{bms}_{+}\oplus\mathfrak{bms}_{0}\oplus\mathfrak{bms}_{-}$, where $\mathfrak{bms}_{\pm}=\bigoplus_{n\in\mathbb{Z}_{\pm}}\Bbbk L_{n}\oplus\Bbbk M_{n}$.

Definition 3.4**.**

Let $h_{L},h_{M},c_{L},c_{M}\in\Bbbk$. Define the one-dimensional $\mathfrak{bms}_{0}\oplus\mathfrak{bms}_{+}$-module

[TABLE]

where

[TABLE]

and $L_{n>0}\ket{h_{L},h_{M},c_{L},c_{M}}=M_{n>0}\ket{h_{L},h_{M},c_{L},c_{M}}=0$. The Verma module of the BMS3 algebra $V(h_{L},h_{M},c_{L},c_{M})$ is the $\mathfrak{bms}$-module induced from $\Bbbk_{h_{L},h_{M},c_{L},c_{M}}$. That is,

[TABLE]

The Verma module $V(h_{L},h_{M},c_{L},c_{M})$ admits a Poincaré–Birkhoff–Witt basis of monomials

[TABLE]

where $n_{k}\geq n_{k-1}\geq\dots\geq n_{1}\geq 1$, $m_{\ell}\geq m_{\ell-1}\geq\dots m_{1}\geq 1$. $V(h_{L},h_{M},c_{L},c_{M})$ is therefore cyclically generated by the action of $\mathfrak{bms}_{-}$ on $\ket{h_{L},h_{M},c_{L},c_{M}}$. As left $U(\mathfrak{bms})$-modules,

[TABLE]

The following proposition describes $V(h_{L},h_{M},c_{L},c_{M})$ as a $\widehat{\mathfrak{b}}$-module.

Proposition 3.5**.**

As $\widehat{\mathfrak{b}}$-modules,

[TABLE]

Proof.

Similar to the proof of Proposition 3.3, with the only relations given by the ideal by which we quotient coming from $P_{0},C_{M}\in\widehat{\mathfrak{b}}$. ∎

Remark 3.6**.**

Although such “almost free” modules are not the most interesting from a representation theoretic point of view, this restriction seems to hint at some interesting physics. There exists a notion that the distinction between closed and open strings blur in the tensionless limit [FMS07, BBP19], one that the preliminary findings of [BCC*+*25b] also corroborate. Given that a $\mathfrak{bms}$ Verma module is one of the possible Hilbert spaces of closed tensionless strings [BBC*+*20], its lack of alteration under restriction to the open string setting (under a certain choice of boundary condition) is suggestive of this fact.

3.3. $\widehat{\mathfrak{b}}$-modules via restriction of massive modules

We consider a class of BMS3 modules which we call massive modules333In the physics literature, the modules we introduce in this subsection are called induced modules, and the term “massive modules” only refers to a subset of them. However, we use the term “massive modules” to refer to all such modules instead, as the term “induced modules” would be too vague in our context. [BO14, BO15, CGOR16], defined as follows.

Definition 3.7**.**

Let $\mathfrak{h}$ denote the subalgebra of $\mathfrak{bms}$ spanned by $\{M_{n}\mid n\in\mathbb{Z}\}$ and $L_{0}$. In other words, $\mathfrak{h}$ is the extension-by-derivation of the abelian ideal $I(0,-1)$ of $\mathfrak{bms}$ spanned by $\{M_{n}\mid n\in\mathbb{Z}\}$, where $\operatorname{ad}_{L_{0}}$ is the derivation by which it is extended. This is summarised by the short exact sequence

[TABLE]

Let $\widehat{\mathfrak{h}}$ be the subalgebra $\mathfrak{h}\oplus\Bbbk C_{L}\oplus\Bbbk C_{M}$. Letting $\mathbf{M},\mathbf{s},c_{L},c_{M}\in\Bbbk$, consider the one-dimensional $\widehat{\mathfrak{h}}$-module

[TABLE]

where

[TABLE]

and $M_{n}\ket{\mathbf{M},\mathbf{s}}=0$ for $n\neq 0$. The $\mathfrak{bms}$-modules $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ induced from $\widetilde{\Bbbk}_{\mathbf{M},\mathbf{s},c_{L},c_{M}}$ are called massive modules [BO14, BO15, CGOR16, BBC*+*20]. That is,

[TABLE]

The massive module $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ admits a PBW basis of monomials

[TABLE]

where $n_{1},\dots,n_{k}\in\mathbb{Z}\setminus\{0\}$ with $n_{k}\geq\dots\geq n_{1}$.

The following result characterises the reducibility of the massive modules in terms of the parameters $\mathbf{M},\mathbf{s},c_{L},c_{M}$. To the best of our knowledge, this result has not been explicitly stated and proven in the literature before.

Theorem 3.8**.**

Let $\mathbf{M},\mathbf{s},c_{L},c_{M}\in\Bbbk$. The $\mathfrak{bms}$-module $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ is irreducible if and only if $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n$.

Proof.

We would like to apply Theorem 2.15. Hence, we first aim to relate $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ to a universal quasi-Whittaker module of $\mathfrak{bms}$ as per Definition 2.12. Consider the ideal $\mathfrak{a}\coloneqq\operatorname{span}\{C_{M},C_{L},M_{k}\mid k\in\mathbb{Z}\}$ of $\mathfrak{bms}$ and define the Whittaker function $\phi\colon\mathfrak{a}\to\Bbbk$ given by

[TABLE]

Letting $W_{\phi}\coloneqq\operatorname{Ind}_{\mathfrak{a}}^{\mathfrak{bms}}\Bbbk_{\phi}$ be the associated universal quasi-Whittaker module, it is easy to see that

[TABLE]

where $J_{\mathbf{s}}\coloneqq U(\mathfrak{bms})(L_{0}-\mathbf{s})\cdot 1_{\phi}$.

Suppose first that $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n$. By setting $n=1$, we see that $\mathbf{M}\neq 0$. From Definition 2.14, we may quickly deduce that the Whittaker annihilator of $\phi$ is $\mathfrak{a}\oplus\Bbbk L_{0}$. Now, Theorem 2.15 implies that $J_{\mathbf{s}}$ is a maximal $\mathfrak{bms}$-submodule of $W_{\phi}$. Therefore, the quotient $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})\cong W_{\phi}/J_{\mathbf{s}}$ is irreducible.

Conversely, assume that $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}=0$ for a positive integer $n$. It is easy to compute that the Whittaker annihilator of $\phi$ contains $\mathfrak{a}\oplus\Bbbk L_{0}\oplus\Bbbk L_{n}\oplus\Bbbk L_{-n}$. Therefore, we see that $L_{n}\ket{\mathbf{M},\mathbf{s}}$ generates a nonzero proper submodule of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$. Thus, in this case, $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ is reducible. ∎

Remark 3.9**.**

By comparing [JZ15, Theorem 2.3] with Theorem 3.8, we see that the massive modules $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ are irreducible under the same conditions as the Verma modules of $\mathfrak{bms}$ from Definition 3.4. This is an unexpected, intriguing similarity between two very different classes of $\mathfrak{bms}$-modules that could potentially be explained by structural features of these modules that are yet to be discovered.

Using the findings from the proof of Theorem 3.8 and another result from [CGL*+*25], it is not difficult to show that $L_{n}\ket{\mathbf{M},\mathbf{s}}$ and $L_{-n}\ket{\mathbf{M},\mathbf{s}}$ together generate a maximal submodule of the massive module $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ when $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}=\mathbf{s}+\frac{n^{2}-1}{24}c_{L}=0$ and $c_{M}\neq 0$.

Corollary 3.10**.**

Let $\mathbf{M},\mathbf{s},c_{L},c_{M}\in\Bbbk$ with $c_{M}\neq 0$, and suppose $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}=\mathbf{s}+\frac{n^{2}-1}{24}c_{L}=0$ for some positive integer $n$. Then the submodule of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ generated by $L_{n}\ket{\mathbf{M},\mathbf{s}}$ and $L_{-n}\ket{\mathbf{M},\mathbf{s}}$ is a maximal submodule of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$.

Proof.

Let $\phi\colon\mathfrak{a}\to\Bbbk$ be as in the proof of Theorem 3.8. Let

[TABLE]

and define $\Phi\colon\widetilde{\mathfrak{a}}\to\Bbbk$ by ${\left.\kern-1.2pt\Phi\vphantom{\big|}\right|_{\mathfrak{a}}}=\phi$, and

[TABLE]

One can compute that $\mathfrak{bms}^{\Phi}=\widetilde{\mathfrak{a}}$. The condition that $\mathbf{s}+\frac{n^{2}-1}{24}c_{L}=0$ guarantees that $\Phi$ is a well-defined Lie algebra homomorphism, since

[TABLE]

Therefore, $\Phi$ extends the quasi-Whittaker function $\phi$ to the entirety of its Whittaker annihilator. By [CGL*+*25, Lemma 3.7], it follows that $\operatorname{Ind}_{\widetilde{\mathfrak{a}}}^{\mathfrak{bms}}\Bbbk_{\Phi}$ is an irreducible $\mathfrak{bms}$-module. It is easy to see that this module is isomorphic to the quotient of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ by its submodule generated by $L_{n}\ket{\mathbf{M},\mathbf{s}}$ and $L_{-n}\ket{\mathbf{M},\mathbf{s}}$, yielding the result. ∎

Remark 3.11**.**

We were informed by Blagoje Oblak that Theorem 3.8 and Corollary 3.10 match the intuitive expectation stemming from his work [BO15] with Glenn Barnich, in which the authors study coadjoint orbits of the BMS3 group and provide classical and quantum mechanical interpretations of these findings in the context of three-dimensional asymptotically flat gravity. It is reassuring that we are able to make this physical intuition mathematically precise.

We now consider the reducibility of the $\mathfrak{b}$-submodule of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ generated by the element $\ket{\mathbf{M},\mathbf{s}}$. The condition governing its reducibility turns out to be exactly the same as the one in Theorem 3.8, and the proof is easier, since $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}$ is a quasi-Whittaker module over $\widehat{\mathfrak{b}}$ on the nose.

Proposition 3.12**.**

Let $\mathbf{M},\mathbf{s},c_{L},c_{M}\in\Bbbk$. The $\widehat{\mathfrak{b}}$-module $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}$ is irreducible if and only if $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n$.

Proof.

Suppose first that $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n.$ Again, setting $n=1$ lets us infer that $\mathbf{M}\neq 0$. Take $\mathfrak{a}\coloneqq\operatorname{span}\{C_{M},P_{k}:k\geq 0\}$, which is an ideal of $\widehat{\mathfrak{b}}$, and define a linear map $\phi\colon\mathfrak{a}\to\Bbbk$ by

[TABLE]

The Whittaker annihilator of $\phi$, denoted $\widehat{\mathfrak{b}}^{\phi}$, is easily computed to be $\widehat{\mathfrak{b}}^{\phi}=\mathfrak{a}$. It now follows immediately from Theorem 2.15 that the $\widehat{\mathfrak{b}}$-module $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}\cong W_{\phi}=\operatorname{Ind}_{\mathfrak{a}}^{\widehat{\mathfrak{b}}}\Bbbk_{\phi}$ is irreducible.

Conversely, assume that $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}=0$ for a positive integer $n$. In this case, $\widehat{\mathfrak{b}}^{\phi}\supseteq\mathfrak{a}\oplus\Bbbk\mathcal{O}_{n}$. Then from Theorem 2.15, we see that $\mathcal{O}_{n}\ket{\mathbf{M},\mathbf{s}}$ generates a nonzero proper submodule of $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}$. Thus, $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}$ is reducible in this case. ∎

Thus, the massive modules of $\mathfrak{bms}$ seem to give rise to irreducible $\widehat{\mathfrak{b}}$-modules more readily than the Verma modules.

We finish our analysis of massive modules by studying the full space $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ as a module over $\widehat{\mathfrak{b}}$. In determining the structure $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ as a $\widehat{\mathfrak{b}}$-module, we assume that $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$ for any positive integer $n$ for the rest of this section, meaning that $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}$ is irreducible. We first introduce two new symbols.

Notation 3.13**.**

For $n\in\mathbb{Z}$, define

[TABLE]

as elements of $\mathfrak{bms}$. Note that $\widehat{P}_{n}$ and $\widehat{\mathcal{O}}_{n}$ are not elements of $\widehat{\mathfrak{b}}$.

By (2.1), a simple computation shows that for any $n,m\in\mathbb{N}$ we have

[TABLE]

Notice that $\{\mathcal{O}_{n},\widehat{\mathcal{O}}_{m},P_{m},\widehat{P}_{n}\mid n\in\mathbb{Z}_{+},m\in\mathbb{N}\}$ is another basis of $\mathfrak{bms}$. Hence, by the PBW theorem, $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ has a basis given by monomials of the form

[TABLE]

where $p_{k}>\dots>p_{1}\geq 1$, $r_{\ell}>\dots>r_{1}\geq 1$ and $q_{1},\dots,q_{k},s_{1},\dots,s_{\ell}\in\mathbb{N}$.

We now prove that the element $\widehat{\mathcal{O}}_{n}\ket{\mathbf{M},\mathbf{s}}$ of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ generates an irreducible $\widehat{\mathfrak{b}}$-submodule, for all $n\in\mathbb{Z}_{+}$.

Proposition 3.14**.**

For any $n\in\mathbb{Z}_{+}$, the $\widehat{\mathfrak{b}}$-module $U(\widehat{\mathfrak{b}})\widehat{\mathcal{O}}_{n}\ket{\mathbf{M},\mathbf{s}}$ is isomorphic to $U(\widehat{\mathfrak{b}})\ket{\mathbf{M},\mathbf{s}}$. Consequently, $U(\widehat{\mathfrak{b}})\widehat{\mathcal{O}}_{n}\ket{\mathbf{M},\mathbf{s}}$ is an irreducible $\widehat{\mathfrak{b}}$-module.

Proof.

It follows from (3.2) that for any $r\in\mathbb{N}$,

[TABLE]

As above, the PBW theorem implies that $U(\widehat{\mathfrak{b}})\widehat{\mathcal{O}}_{n}\ket{\mathbf{M},\mathbf{s}}$ has a basis given by

[TABLE]

where $p_{k}>\dots>p_{1}\geq 1$, and $q_{1},\dots,q_{k}\in\mathbb{N}$. Therefore, we see that the map

[TABLE]

where $p_{k}>\dots>p_{1}\geq 1$ and $q_{1},\dots,q_{k}\in\mathbb{N}$, is an isomorphism of $\widehat{\mathfrak{b}}$-modules.

The final sentence follows from Proposition 3.12 upon recalling that we are assuming that $\mathbf{M}+\frac{n^{2}-1}{24}c_{M}\neq 0$. ∎

As we will soon see, $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ decomposes into a direct sum of two $\widehat{\mathfrak{b}}$-submodules. We define these submodules below.

Definition 3.15**.**

Let $\widehat{V}$ be the $\widehat{\mathfrak{b}}$-submodule of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ generated by

[TABLE]

and let $\widehat{W}$ be the $\widehat{\mathfrak{b}}$-submodule of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ generated by

[TABLE]

The following result shows that the action of $P_{r}$ on $\widehat{\mathcal{O}}^{s_{\ell}}_{r_{\ell}}\dots\widehat{\mathcal{O}}^{s_{1}}_{r_{1}}\ket{\mathbf{M},\mathbf{s}}$ preserves the parity of $\sum s_{i}$.

Proposition 3.16**.**

For any $r,s_{1},\ldots,s_{\ell}\in\mathbb{N}$ and $r_{\ell}>\dots>r_{1}\geq 1$, the abelian part of the BMS3 algebra has the following action on $\widehat{\mathcal{O}}^{s_{\ell}}_{r_{\ell}}\dots\widehat{\mathcal{O}}^{s_{1}}_{r_{1}}\ket{\mathbf{M},\mathbf{s}}$:

[TABLE]

Proof.

We prove the result by induction on $d\coloneqq\sum_{i=1}^{\ell}s_{i}$. It is not difficult to check that the two equalities hold for $d=0,1$. Now assume that $d>1$ and the two equalities hold for all $0\leq d^{\prime}<d$. Consider the first equality for the case $d=\sum_{i=1}^{\ell}s_{i}$. We have

[TABLE]

The induction hypothesis implies that $\left((r-r_{\ell})\widehat{P}_{r+r_{\ell}}+(r+r_{\ell})\widehat{P}_{r-r_{\ell}}\right)\widehat{\mathcal{O}}^{s_{\ell}-1}_{r_{\ell}}\dots\widehat{\mathcal{O}}^{s_{1}}_{r_{1}}\ket{\mathbf{M},\mathbf{s}}$ belongs to

[TABLE]

and that $P_{r}\widehat{\mathcal{O}}^{s_{\ell}-1}_{r_{\ell}}\dots\widehat{\mathcal{O}}^{s_{1}}_{r_{1}}\ket{\mathbf{M},\mathbf{s}}-\delta_{0,r}2{\bf M}\widehat{\mathcal{O}}^{s_{\ell}-1}_{r_{\ell}}\dots\widehat{\mathcal{O}}^{s_{1}}_{r_{1}}\ket{\mathbf{M},\mathbf{s}}$ belongs to

[TABLE]

The first equality follows. The second equality can be proved similarly. ∎

This naturally leads to the decomposition of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ given by the following corollary.

Corollary 3.17**.**

As a $\widehat{\mathfrak{b}}$-module, $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ is decomposable. More precisely, we have

[TABLE]

Proof.

From Proposition 3.16 and the PBW theorem, we may deduce that $\widehat{V}$ and $\widehat{W}$ have bases given by

[TABLE]

respectively. Comparing this with the PBW basis of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ given by (3.3) leads to the result. ∎

We finish our discussion of $\widetilde{V}(\mathbf{M},\mathbf{s},c_{L},c_{M})$ as a $\widehat{\mathfrak{b}}$-module with the following conjecture.

Conjecture 3.18**.**

The $\widehat{\mathfrak{b}}$-modules $\widehat{V}$ and $\widehat{W}$ are indecomposable.

3.4. The $\mathcal{O}$-module $\Omega(\lambda,a)$

We finish this section by restricting one final class of Virasoro modules to $\mathcal{O}$, first defined in [LZ14]. For any pair $(a,\lambda)\in\Bbbk\times\Bbbk^{*}$, one can define a Virasoro algebra module structure on $\Bbbk[X]$ by

[TABLE]

The module is irreducible if and only if $a\neq 0$, and $\Omega(\lambda,0)$ has a unique irreducible submodule which is isomorphic to $\Omega(\lambda,1)$ [LZ10, Section 4.3]. Restricting the above action to the subalgebra $\mathcal{O}$ of $\operatorname{Vir}$, the $\mathcal{O}$-module structure on $\Bbbk[X]$ is given by

[TABLE]

We still denote the corresponding $\mathcal{O}$-module by $\Omega(\lambda,a)$.

Proposition 3.19**.**

Suppose $\lambda\in\Bbbk^{*}$ and $a\in\Bbbk$. If $\lambda\neq\pm 1$ and $a\neq 0$, then the $\mathcal{O}$-module $\Omega(\lambda,a)$ is irreducible.

To prove Proposition 3.19, we need the following lemma from [TZ13].

Lemma 3.20** ([TZ13, Lemma 2]).**

Let $\lambda_{1},\lambda_{2},\dots,\lambda_{m}\in\Bbbk$, $s_{1},s_{2},\dots,s_{m}\in\mathbb{Z}_{+}$ with $s_{1}+\dots+s_{m}=s$. Define a sequence of functions on $\mathbb{Z}$ as follows: $f_{1}(n)=\lambda_{1}^{n},f_{2}(n)=n\lambda_{1}^{n},\dots,f_{s_{1}}(n)=n^{s_{1}-1}\lambda_{1}^{n},f_{s_{1}+1}(n)=\lambda_{2}^{n},\dots,f_{s_{1}+s_{2}}(n)=n^{s_{2}-1}\lambda_{2}^{n},\dots,f_{s}(n)=n^{s_{m}-1}\lambda_{m}^{n}$. Let $\mathcal{M}=(y_{pq})$ be the $s\times s$ matrix with $y_{pq}=f_{q}(p-1)$, $q=1,2,\dots,s$, $p=u+1,u+2,\dots,u+s$ where $u\in\mathbb{N}$. Then

[TABLE]

where $s_{j}!!=s_{j}!\times(s_{j}-1)!\times\dots\times 2!\times 1!$ with $0!!=1$, for convenience.

In particular, we have the following.

Lemma 3.21**.**

Suppose $\mu\in\Bbbk$ and $\mu\neq 0,1$. For $r\in\mathbb{Z}_{+}$ and a fixed $u\in\mathbb{N}$ let $k_{j}=u+j,j=1,2,\dots,2r$. Letting

[TABLE]

then $\tau(\mu,r,u)\neq 0$.

Proof.

Taking $m=2$, $\lambda_{1}=1,\lambda_{2}=\mu$ and $s_{1}=s_{2}=r$ in Lemma 3.20 we see $\tau(\mu,r,u)=\det\big((y_{pq})_{2r\times 2r}\big)$ with $y_{pq}=f_{q}(p-1)$ and $q=1,2,\dots,2r,p=u+1,u+2,\dots,u+2r$. So

[TABLE]

as desired. This completes the proof. ∎

As the following result shows, the key difference between the cases where $\lambda=\pm 1$ or $a=0$ and the other cases is whether $\Omega(\lambda,a)$ is generated by $1$.

Lemma 3.22**.**

If $\lambda\neq\pm 1$ and $a\neq 0$, then the $\mathcal{O}$-module $\Omega(\lambda,a)$ is generated by $1$.

Proof.

It is sufficient to show that $X^{n}\in U(\mathcal{O})\cdot 1$ for any $n\in\mathbb{N}.$ We prove this by induction on $n$, the base case $n=0$ being trivial. For the induction step, let $n>0$ and assume $X^{n^{\prime}}\in U(\mathcal{O})\cdot 1$ for any $0\leq n^{\prime}<n$. We aim to show that $X^{n}\in U(\mathcal{O})\cdot 1.$ Since $X^{n-1}\in U(\mathcal{O})\cdot 1$, it follows from (3.4) that

[TABLE]

for some $g(X)\in\Bbbk[X]$. Note that $g(X)\in U(\mathcal{O})\cdot 1$ by the induction hypothesis, since $\deg(g(X))\leq n-1$. Therefore,

[TABLE]

Since $\lambda\neq\pm 1$, there always exists $k\geq 1$ such that $(\lambda^{2k}-1)\neq 0$, which implies that $X^{n}\in U(\mathcal{O})\cdot 1$. Thus, $U(\mathcal{O})\cdot 1=\Omega(\lambda,a)$. This completes the proof. ∎

Now we can prove Proposition 3.19.

Proof of Proposition 3.19..

To show that $\Omega(\lambda,a)$ is simple is enough to show that any nonzero submodule of $\Omega(\lambda,a)$ is the module $\Omega(\lambda,a)$ itself. Let $M$ be a nonzero submodule of $\Omega(\lambda,a)$. To show $M=\Omega(\lambda,a)$ is, by Lemma 3.22, to show $1\in M$. To the contrary, assume that $1\notin M$. Let $f(X)\in M$ be a nonzero element with minimal degree $r>0$. Denote $f(X)=\sum_{i=0}^{r}c_{i}X^{i}$ with $c_{i}\in\Bbbk$ and $c_{r}\neq 0$. Then by (3.4) we have

[TABLE]

for all $k\in\mathbb{Z}_{+}$, where $\varphi_{i}(X),\psi_{i}(X)\in\Bbbk[X]$ and $\varphi_{r+1}(X)=ac_{r}$ and $\psi_{r+1}(X)=(-1)^{r}ac_{r}$ are nonzero constants. Let $\mu=\lambda^{2}$, $k_{j}=j$, where $1\leq j\leq 2(r+2)$ and $\mu\neq 0,1$. By Lemma 3.21 we know that $\tau(\mu,r+2,0)\neq 0$, which implies that $\varphi_{i}(X),\psi_{i}(X)\in M$ for all $0\leq i\leq r+1$. In particular, $\varphi_{r+1}(X),\psi_{r+1}(X)\in M$, contrary to the choice of $f(X)$. Therefore, $1\in M$. So $M=\Omega(\lambda,a)$ and $\Omega(\lambda,a)$ is a simple $\mathcal{O}$-module. This completes the proof. ∎

Finally, we consider the case $\lambda=\pm 1.$

Proposition 3.23**.**

If $\lambda=\pm 1$ and $a\neq 0$, then the $\mathcal{O}$-module $\Omega(\lambda,a)$ carries a filtration of submodules of $\Omega(\lambda,a)$ as follows:

[TABLE]

where

[TABLE]

Proof.

We only need to show that each $\Omega(\lambda,a)_{n}$ is a submodule of $\Omega(\lambda,a)$, which follows from the fact that for any $k\geq 1$ and any nonzero $f(X)\in\Bbbk[X]$,

[TABLE]

has degree $\leq\deg(f(X))$ for $\deg(f(X+k)-f(X-k))=\deg(f(X))-1$ if $\deg(f(X))\geq 1$. ∎

Remark 3.24**.**

Since $\Omega(\lambda,0)$ as a $\operatorname{Vir}$-module has a submodule $X\Bbbk[X]$, which is isomorphic to $\Omega(\lambda,1)$, we see that $\Omega(\lambda,0)$ as a $\mathcal{O}$-module also has a submodule $X\Bbbk[X]\cong\Omega(\lambda,1)$. If $\lambda\neq\pm 1$, then, by Proposition 3.19, $X\Bbbk[X]$ is a simple $\mathcal{O}$-module; and if $\lambda=\pm 1$, then, by Proposition 3.23, $\Omega(\lambda,0)$ as an $\mathcal{O}$-module has a filtration of submodules:

[TABLE]

4. Alternate basis of the BCCA

Our next goal is to construct a decreasing filtration on $\mathcal{O}$ and $\mathfrak{b}$, which will allow us to define and study Whittaker modules for these Lie algebras as per Lemma 2.11. Unfortunately, the basis of $\mathfrak{b}$ from Section 2 makes it difficult to construct the desired filtration. Therefore, in this section we present an alternate basis for $\mathcal{O}$ and $\mathfrak{b}$, from which the decreasing filtration becomes manifest.

4.1. Subalgebras of the Witt algebra

Since $\mathfrak{b}=\mathcal{O}\ltimes\mathcal{P}$, where $\mathcal{O}$ is a subalgebra of $\mathcal{W}$, we now spend some time outlining some general facts about subalgebras of the Witt algebra. Note that $\mathcal{W}$ is naturally a $\Bbbk[t,t^{-1}]$-module, and thus the most natural subalgebras of $\mathcal{W}$ are those which are also $\Bbbk[t,t^{-1}]$-submodules.

Definition 4.1**.**

Given $f\in\Bbbk[t,t^{-1}]$, define

[TABLE]

so that $f\mathcal{W}$ consists of all derivations of $\Bbbk[t,t^{-1}]$ which are divisible by $f$. Given $f\in\Bbbk[t]$, we define $f\mathbb{W}_{1}$ in a similar way. The Lie subalgebras $f\mathcal{W}$ and $f\mathbb{W}_{1}$ are known as submodule-subalgebras of $\mathcal{W}$ and $\mathbb{W}_{1}$.

Note that the bracket in $f\mathcal{W}$ is given by

[TABLE]

for all $g,h\in\Bbbk[t,t^{-1}]$. The same formula holds in $f\mathbb{W}_{1}$, except we require that $f,g,h\in\Bbbk[t]$.

Example 4.2**.**

Letting $f\coloneqq t^{2}-4$, we define $\mathfrak{f}\coloneqq f\mathbb{W}_{1}$. Then $\mathfrak{f}$ is spanned by

[TABLE]

where $n\geq 1$. The bracket in $\mathfrak{f}$ is

[TABLE]

In fact, all subalgebras of $\mathcal{W}$ and $\mathbb{W}_{1}$ of finite codimension are “essentially” submodule-subalgebras, in a sense that is made precise in the next result.

Proposition 4.3** ([PS23, Proposition 3.2.7]).**

Let $\mathfrak{g}$ be a subalgebra of $\mathcal{W}$ of finite codimension. Then there exist $f\in\Bbbk[t,t^{-1}]\setminus\{0\}$ and $n\in\mathbb{N}$ such that

[TABLE]

The analogous result for finite codimension subalgebras of $\mathbb{W}_{1}$ is also true with a nearly identical proof, although this is not explicitly proved in [PS23].

However, the Lie algebra $\mathcal{O}$ is a subalgebra of $\mathcal{W}$ of infinite codimension, so Proposition 4.3 does not apply. Thankfully, there is an analogous classification of subalgebras of $\mathbb{W}_{1}$ of infinite codimension [BB25, Theorem 2.8]. Although the result classifies subalgebras of $\mathbb{W}_{1}$, not of $\mathcal{W}$, the same ideas and techniques can be applied to study subalgebras of $\mathcal{W}$. To state the result, we must first introduce the subalgebras of $\mathbb{W}_{1}$ which play the role of the submodule-subalgebras in the infinite codimension setting. Following [Buz23, BB25], we introduce the notions of the set and field of ratios associated to a subalgebra of $\mathcal{W}$.

Definition 4.4**.**

Let $\mathfrak{g}$ be a Lie subalgebra of $\mathcal{W}$. The set of ratios of $\mathfrak{g}$ is

[TABLE]

The field of ratios $F(\mathfrak{g})$ of $\mathfrak{g}$ is the subfield of $\Bbbk(t)$ generated by $R(\mathfrak{g})$.

The next result gives one of the most important reasons why $F(\mathfrak{g})$ is a useful tool to study $\mathfrak{g}$.

Lemma 4.5**.**

Let $\mathfrak{g}$ be a subalgebra of $\mathcal{W}$. Then $\mathfrak{g}\subseteq\operatorname{Der}(F(\mathfrak{g}))$.

Proof.

Let $u\partial,v\partial,w\partial\in\mathfrak{g}$ with $v\neq 0$, where $u,v,w\in\Bbbk[t,t^{-1}]$, so that $\frac{u}{v},\frac{w}{v}\in R(\mathfrak{g})$. We claim that $w\partial(\frac{u}{v})\in F(\mathfrak{g})$, which means that elements of $\mathfrak{g}$ map $R(\mathfrak{g})$ to $F(\mathfrak{g})$. Since $F(\mathfrak{g})$ is generated by $R(\mathfrak{g})$ as a field, it will then follow that all derivations in $\mathfrak{g}$ preserve $F(\mathfrak{g})$.

We have

[TABLE]

Certainly, $\frac{w}{v}\in R(\mathfrak{g})$ by definition. Note that $[v\partial,u\partial]=(v\partial(u)-u\partial(v))\partial\in\mathfrak{g}$, so we also have $\frac{v\partial(u)-u\partial(v)}{v}\in R(\mathfrak{g})$. It follows that $w\partial(\frac{u}{v})\in F(\mathfrak{g})$, as claimed. ∎

As the next example illustrates, it is often useful to consider $\mathfrak{g}$ as a subalgebra of $\operatorname{Der}(F(\mathfrak{g}))$ rather than as a subalgebra of $\mathcal{W}$, at least when $\mathfrak{g}$ has infinite codimension in $\mathcal{W}$.

Example 4.6**.**

Consider the subalgebra $\mathfrak{g}=\operatorname{span}\{L_{2n}\mid n\in\mathbb{Z}\}$. It is easy to see that $\mathfrak{g}$ is a subalgebra of $\mathcal{W}$ of infinite codimension and that $F(\mathfrak{g})=R(\mathfrak{g})=\Bbbk(t^{2})$, since $\mathfrak{g}=\Bbbk[t^{2},t^{-2}]t\partial$. Thus, we see that $\mathfrak{g}$ is the subalgebra $\operatorname{Der}(\Bbbk[t^{2},t^{-2}])$ of $\operatorname{Der}(\Bbbk(t^{2}))$. Therefore, when we view $\mathfrak{g}$ as a subalgebra of $\operatorname{Der}(F(\mathfrak{g}))$, we forget about the basis elements $L_{2n+1}$ for all $n\in\mathbb{Z}$, which are not relevant when studying Lie algebraic properties of $\mathfrak{g}$.

Since $F(\mathfrak{g})\subseteq\Bbbk(t)$, Lüroth’s theorem implies that $F(\mathfrak{g})=\Bbbk(s)$ for some $s\in\Bbbk(t)$, so Lemma 4.5 implies that $\mathfrak{g}\subseteq\operatorname{Der}(\Bbbk(s))$. In [Buz23, Theorem 5.3], it was shown that $s$ can be chosen to be a polynomial when $\mathfrak{g}\subseteq\mathbb{W}_{1}$. In fact, even more is true: $\mathfrak{g}$ is contained in $\operatorname{Der}(\Bbbk[s])$ with finite codimension.

Theorem 4.7** ([BB25, Theorem 2.8]).**

Let $\mathfrak{g}$ be an infinite-dimensional subalgebra of $\mathbb{W}_{1}$. Then there exists $s\in\Bbbk[t]$ such that $F(\mathfrak{g})=\Bbbk(s)$, and $\mathfrak{g}$ has finite codimension in $\operatorname{Der}(\Bbbk[s])$. In particular, there exists $g(s)\in\Bbbk[s]$ such that

[TABLE]

where $\partial_{s}\in\operatorname{Der}(\Bbbk[s])$ is the unique derivation of $\Bbbk[s]$ such that $\partial_{s}(s)=1$, so that $\operatorname{Der}(\Bbbk[s])=\Bbbk[s]\partial_{s}$.

Let $\mathfrak{g}\subseteq\mathbb{W}_{1}$, $s\in\Bbbk[t]$, and $\partial_{s}$ be as in Theorem 4.7. Certainly, the Lie algebra $\operatorname{Der}(\Bbbk[s])$ is isomorphic to $\mathbb{W}_{1}$, so Theorem 4.7 says that $\mathfrak{g}$ is isomorphic to a subalgebra of $\mathbb{W}_{1}$ of finite codimension. The derivation $\partial_{s}\in\operatorname{Der}(\Bbbk[s])$ can be uniquely extended to a derivation of $\Bbbk(t)$: we have $\partial_{s}=\frac{1}{s^{\prime}}\partial$, where $s^{\prime}\coloneqq\partial(s)$ is the derivative of $s$ with respect to $t$. This allows us to embed $\operatorname{Der}(\Bbbk[s])$ into $\operatorname{Der}(\Bbbk(t))$ as follows:

[TABLE]

Of course, $\mathbb{W}_{1}=\Bbbk[t]\partial$ is also contained in $\operatorname{Der}(\Bbbk(t))$. Therefore, Theorem 4.7 implies that $\mathfrak{g}\subseteq\mathbb{W}_{1}\cap\operatorname{Der}(\Bbbk[s])$, where the intersection is taken in $\operatorname{Der}(\Bbbk(t))$.

Although Theorem 4.7 only considers subalgebras of $\mathbb{W}_{1}$, the same ideas can be applied to subalgebras of $\mathcal{W}$. To that end, we introduce the following definition.

Notation 4.8**.**

If $s\in\Bbbk[t,t^{-1}]$, we define $L(s)\coloneqq\mathcal{W}\cap\operatorname{Der}(\Bbbk[s])$.

The Lie algebra $L(s)$ can be described as follows.

Lemma 4.9** ([BB25, Lemma 2.7]).**

Let $s\in\Bbbk[t,t^{-1}]$, and let $g_{s}\in\Bbbk[t]$ be the unique monic polynomial of minimal degree such that $s^{\prime}g_{s}\in\Bbbk[s]$. Letting $f_{s}\in\Bbbk[t]$ such that $s^{\prime}g_{s}=f_{s}(s)$, we have $L(s)=\Bbbk[s]g_{s}\partial$.

Remark 4.10**.**

Although the condition $s^{\prime}g_{s}\in\Bbbk[s]$ might seem unusual, it is precisely the necessary condition for $g_{s}\partial$ to be a derivation of $\Bbbk[s]$. Furthermore, it is not obvious that such a polynomial $g_{s}$ exists for every $s\in\Bbbk[t,t^{-1}]$; this is proved in [Buz23, Proposition 4.13].

We can also describe $L(s)$ as a subalgebra of $\operatorname{Der}(\Bbbk[s])=\Bbbk[s]\partial_{s}$ (in other words, viewing it as a Lie algebra of derivations of $\Bbbk[s]$ instead of $\Bbbk[t,t^{-1}]$). Since $\partial=s^{\prime}\partial_{s}$, it follows that

[TABLE]

Therefore, $L(s)$ is the subalgebra $\Bbbk[s]f_{s}(s)\partial_{s}$ of $\operatorname{Der}(\Bbbk[s])$. By changing variables $s\mapsto t$, we see that $L(s)\cong\Bbbk[t]f_{s}\partial=f_{s}\mathbb{W}_{1}$, so $L(s)$ is isomorphic to a subalgebra of $\mathbb{W}_{1}$ of finite codimension. We now give an example to illustrate the construction.

Example 4.11** ([Buz23, Example 4.16]).**

If we let $s=t^{3}+3t$ then $g_{s}=(t^{2}+1)(t^{2}+4)$ and $f_{s}=3(t^{2}+4)$, since $s^{\prime}g_{s}=3(s^{2}+4)\in\Bbbk[s]$. Therefore, $L(s)=\Bbbk[t^{3}+3t](t^{2}+1)(t^{2}+4)\partial$ is a Lie subalgebra of $\mathcal{W}$. As a subalgebra of $\operatorname{Der}(\Bbbk[s])$, we have $L(s)=\Bbbk[s](s^{2}+4)\partial_{s}$. Changing variables $s\mapsto t$, we see that

[TABLE]

so $L(s)$ is isomorphic to the subalgebra $(t^{2}+4)\mathbb{W}_{1}$ of $\mathbb{W}_{1}$ of codimension 2.

We summarise this discussion below.

Proposition 4.12** ([Buz23, Lemma 4.12, Proposition 4.13]).**

Let $s\in\Bbbk[t,t^{-1}]$, and let $f_{s},g_{s}\in\Bbbk[t]$ be as in Lemma 4.9. Then

[TABLE]

where the isomorphism comes from the change of variables $s\mapsto t$.

4.2. Change of basis for $\mathcal{O}$

We now apply the ideas of Subsection 4.1 to the Lie subalgebra $\mathcal{O}$ of $\mathcal{W}$ to construct a new basis of $\mathcal{O}$ which will allow us to define a descending filtration for the BCCA. In fact, we work in a slightly more general setting: $\mathcal{O}$ is part of a family of Lie subalgebras of $\mathcal{W}$. It is not much more difficult to consider the entire family for the proofs in this section, so we work in this level of generality.

Definition 4.13**.**

For $\lambda\in\Bbbk^{*}$ and $n\in\mathbb{Z}$, define $\mathcal{O}^{(\lambda)}_{n}\coloneqq L_{n}-\lambda^{n}L_{-n}$. Notice that $\mathcal{O}^{(\lambda)}_{0}=0$ and $\mathcal{O}^{(\lambda)}_{-n}=-\lambda^{-n}\mathcal{O}^{(\lambda)}_{n}$ for $n\geq 1$. Define $\mathcal{O}(\lambda)\coloneqq\operatorname{span}\{\mathcal{O}^{(\lambda)}_{n}\mid n\geq 1\}$.

Of course $\mathcal{O}(1)=\mathcal{O}$. On the other hand, it is not immediately clear that $\mathcal{O}(\lambda)$ is closed under the Lie bracket of $\mathcal{W}$ for $\lambda\neq 1$. We prove this next.

Lemma 4.14**.**

Let $\lambda\in\Bbbk^{*}$. Then $\mathcal{O}(\lambda)$ is a Lie algebra with the following Lie bracket:

[TABLE]

for all $n,m\geq 1$. In particular, $\mathcal{O}(\lambda)$ is generated by $\mathcal{O}^{(\lambda)}_{1}$ and $\mathcal{O}^{(\lambda)}_{2}$.

Proof.

We compute the bracket $[\mathcal{O}^{(\lambda)}_{n},\mathcal{O}^{(\lambda)}_{m}]$ as follows:

[TABLE]

The final sentence follows by an easy induction. ∎

Remark 4.15**.**

One might ask if we can get different subalgebras of the Witt algebra by combining $L_{n}$ and $L_{-n}$ differently to the way it is done in $\mathcal{O}(\lambda)$. In other words, can we generate different Lie subalgebras of $\mathcal{W}$ with elements $L_{1}-\lambda L_{-1}$ and $L_{2}-\mu L_{-2}$, where $\lambda,\mu\in\Bbbk^{*}$? This question is answered in Appendix A: if $\mu\neq\lambda^{2}$, then these two elements generate the entire Witt algebra. As a result, the subalgebras $\mathcal{O}(\lambda)$ are the unique family of proper subalgebras of $\mathcal{W}$ generated by elements of the form $L_{1}-\lambda L_{-1}$ and $L_{2}-\mu L_{-2}$ for some $\lambda,\mu\in\Bbbk^{*}$.

We now aim to describe an alternative basis $u_{n}^{(\lambda)}$ for $\mathcal{O}(\lambda)$. To achieve this, we analyse the structure of $\mathcal{O}(\lambda)$ following the methods of Subsection 4.1. In terms of polynomials, we have $\mathcal{O}^{(\lambda)}_{1}=-(t^{2}-\lambda)\partial$, and

[TABLE]

Dividing $\mathcal{O}^{(\lambda)}_{2}$ by $\mathcal{O}^{(\lambda)}_{1}$, it follows that

[TABLE]

We therefore define $s_{\lambda}\coloneqq t+\lambda t^{-1}$, so that $\Bbbk(s_{\lambda})\subseteq F(\mathcal{O}(\lambda))$. In fact, as we show next, we have $R(\mathcal{O}(\lambda))=F(\mathcal{O}(\lambda))=\Bbbk(s_{\lambda})$ and $\mathcal{O}(\lambda)=L(s_{\lambda})$, where this notation is defined in Definition 4.4 and Notation 4.8.

Theorem 4.16**.**

Let $\lambda\in\Bbbk^{*}$, and define $\mathfrak{f}_{\lambda}\coloneqq f_{s_{\lambda}}\mathbb{W}_{1}$, where $f_{s_{\lambda}}=t^{2}-4\lambda$. Then

[TABLE]

where the isomorphism comes from the change of variables $s_{\lambda}\mapsto t$. Consequently, $\mathcal{O}(\lambda)$ has another basis $\{u^{(\lambda)}_{n}\mid n\geq 1\}$ defined as follows: for $n\geq 1$, let

[TABLE]

The Lie bracket of $\mathcal{O}(\lambda)$ with this basis is given by

[TABLE]

for all $n,m\geq 1$.

Proof.

We adopt the notation $g_{s_{\lambda}}$ and $f_{s_{\lambda}}$ from Lemma 4.9. It is not difficult to see that $g_{s_{\lambda}}=t^{2}-\lambda$, since

[TABLE]

Consequently, $f_{s_{\lambda}}=t^{2}-4\lambda$. It follows from Proposition 4.12 that

[TABLE]

Note that $\mathcal{O}^{(\lambda)}_{1}=-(t^{2}-\lambda)\partial\in L(s_{\lambda})$ and

[TABLE]

Since $\mathcal{O}^{(\lambda)}_{1}$ and $\mathcal{O}^{(\lambda)}_{2}$ generate $\mathcal{O}(\lambda)$ by Lemma 4.14, this implies that $\mathcal{O}(\lambda)\subseteq L(s_{\lambda})$.

For the other inclusion $L(s_{\lambda})\subseteq\mathcal{O}(\lambda)$, we will show that $L(s_{\lambda})$ is also generated by $\mathcal{O}^{(\lambda)}_{1}$ and $\mathcal{O}^{(\lambda)}_{2}$. To that end, noting that $\{u^{(\lambda)}_{n}\mid n\geq 1\}$ is a basis for $L(s_{\lambda})$, we compute the bracket $[u^{(\lambda)}_{n},u^{(\lambda)}_{m}]$. Letting $\mathfrak{f}_{\lambda}\coloneqq f_{s_{\lambda}}\mathbb{W}_{1}=(t^{2}-4\lambda)\mathbb{W}_{1}$, the change of variables $s_{\lambda}\mapsto t$ yields an isomorphism $L(s_{\lambda})\cong\mathfrak{f}_{\lambda}$, as in Proposition 4.12. Noting that

[TABLE]

we see that

[TABLE]

under the change of variables $s_{\lambda}\mapsto t$. We have

[TABLE]

and therefore,

[TABLE]

An easy induction shows that $\mathcal{O}^{(\lambda)}_{1}=u^{(\lambda)}_{1}$ and $\mathcal{O}^{(\lambda)}_{2}=u^{(\lambda)}_{2}$ generate $L(s_{\lambda})$, which concludes the proof. ∎

Remark 4.17**.**

Theorem 4.16 implies that $R(\mathcal{O}(\lambda))=F(\mathcal{O}(\lambda))=\Bbbk(s_{\lambda})$, although a priori it is not obvious that $R(\mathcal{O}(\lambda))$ is a field.

Example 4.18**.**

The Lie algebra $\mathcal{O}(-1)$ is spanned by $L_{n}+(-1)^{n+1}L_{-n}$ and is isomorphic to $(t^{2}+4)\mathbb{W}_{1}$.

Using Theorem 4.16, it is now easy to see that $\mathcal{O}(\lambda)$ is never simple.

Corollary 4.19**.**

For all $\lambda\in\Bbbk^{*}$, the Lie algebra $\mathcal{O}(\lambda)$ is not perfect, and therefore is not simple.

Proof.

The Lie algebra $\mathfrak{f}_{\lambda}$ is not perfect: this is because the derived subalgebra of $f^{(\lambda)}$ is

[TABLE]

by (4.1). The result now follows by the isomorphism $\mathcal{O}(\lambda)\cong\mathfrak{f}_{\lambda}$ from Theorem 4.16. ∎

Remark 4.20**.**

Corollary 4.19 implies that $\mathfrak{b}$ is not perfect, meaning it does not have a universal central extension [Wei94, Theorem 7.9.2]. Nonetheless, it would be interesting to compute the centrally closed version of $\mathfrak{b}$, which would be the central extension of $\mathfrak{b}$ with trivial second cohomology group (i.e., does not admit any more non-trivial central extensions). From a physics perspective, this would tell us the possible “central charges” that a quantum field theory with $\mathfrak{b}$ symmetry could admit. It remains to be seen whether this centrally closed Lie algebra is indeed $\widehat{\mathfrak{b}}$.

As we aim to show next, the Lie algebras $\mathcal{O}(\lambda)$ are often isomorphic to each other for different values of $\lambda\in\Bbbk^{*}$. Recalling that $\mathcal{O}(\lambda)\cong\mathfrak{f}_{\lambda}=(t^{2}-4\lambda)\mathbb{W}_{1}$, it suffices to consider when two subalgebras of $\mathbb{W}_{1}$ of finite codimension are isomorphic. In fact, this can only happen if the isomorphism is induced by an automorphism of $\mathbb{W}_{1}$ [Buz24]. The automorphisms of $\mathbb{W}_{1}$ are well-known [Rud86].

Theorem 4.21** ([Buz24, Theorem 4.1 and Corollary 4.12]).**

Let $\mathfrak{g}$ and $\mathfrak{h}$ be subalgebras of $\mathbb{W}_{1}$ of finite codimension and suppose there is an isomorphism $\varphi:\mathfrak{g}\to\mathfrak{h}$. Then $\varphi$ extends to an automorphism of $\mathbb{W}_{1}$. In particular, $g\mathbb{W}_{1}\cong h\mathbb{W}_{1}$ if and only if there exist $\alpha,\gamma\in\Bbbk^{*}$ and $\beta\in\Bbbk$ such that $g(z)=\gamma h(t)$, where $z=\alpha t+\beta$ and $g,h\in\Bbbk[t]$.

We are now ready to characterise when $\mathcal{O}(\lambda)\cong\mathcal{O}(\mu)$.

Proposition 4.22**.**

Let $\lambda,\mu\in\Bbbk^{*}$. Then $\mathcal{O}(\lambda)\cong\mathcal{O}(\mu)$ if and only if $\sqrt{\frac{\lambda}{\mu}}\in\Bbbk$.

Proof.

By Theorem 4.16, we have $\mathcal{O}(\lambda)\cong\mathfrak{f}_{\lambda}=(t^{2}-4\lambda)\mathbb{W}_{1}$ and $\mathcal{O}(\mu)\cong\mathfrak{f}_{\mu}=(t^{2}-4\mu)\mathbb{W}_{1}$. Therefore, we will prove the result by showing that $\mathfrak{f}_{\lambda}\cong\mathfrak{f}_{\mu}$ if and only if $\sqrt{\frac{\mu}{\lambda}}\in\Bbbk$.

First, assume that $\alpha\coloneqq\sqrt{\frac{\lambda}{\mu}}\in\Bbbk$. Letting $z\coloneqq\alpha t$, we have

[TABLE]

By Theorem 4.21, the change of variables $t\mapsto\alpha t$ induces an isomorphism $\mathfrak{f}_{\lambda}\cong\mathfrak{f}_{\mu}$.

Conversely, assume that $\mathfrak{f}_{\lambda}\cong\mathfrak{f}_{\mu}$. By Theorem 4.21, there exist $\alpha,\gamma\in\Bbbk^{*}$ and $\beta\in\Bbbk$ such that $f_{s_{\lambda}}(z)=\gamma f_{s_{\mu}}(t)$, where $z=\alpha t+\beta$. Therefore,

[TABLE]

It follows that $\gamma=\alpha^{2}$, $\beta=0$, and $\lambda=\gamma\mu=\alpha^{2}\mu$. Therefore, $\alpha=\sqrt{\frac{\lambda}{\mu}}$ must be an element of $\Bbbk$. ∎

With $\mathcal{O}(1)=\mathcal{O}$, the next result follows immediately from Proposition 4.22.

Corollary 4.23**.**

If $\Bbbk$ is algebraically closed (for example, if $\Bbbk=\mathbb{C}$), then $\mathcal{O}(\lambda)\cong\mathcal{O}$ for all $\lambda\in\Bbbk$. If $\Bbbk=\mathbb{R}$, then

[TABLE]

but $\mathcal{O}(-1)\not\cong\mathcal{O}$. ∎

Remark 4.24**.**

Corollary 4.23 is intriguing in the context in which the BCCA was discovered [BCC*+*25b]. Specifically, we have shown that the subalgebras $\mathcal{O}=\operatorname{span}\{L_{n}-L_{-n}\mid n\geq 1\}$ and $\mathcal{O}(-1)=\operatorname{span}\{L_{n}+(-1)^{n+1}L_{-n}\}$ are isomorphic as complex Lie algebras but not as real ones. In the realisation of the BMS3 algebra given by [BCC*+*25b, Equation (4)] in terms of vector fields which preserve the Carrollian structure on cylinder (see [CSZ24, Section 2.1]), we have

[TABLE]

where $\sigma$ and $\tau$ are the coordinates of a cylinder. The Lie algebra $\mathcal{O}$ was the constructed as the subalgebra of the Witt algebra spanned by $\{L_{n}\mid n\in\mathbb{Z}\}$ which leaves the choice of boundary conditions $\sigma=0,\pi$ invariant [BCC*+*25b, Equation (5)]

[TABLE]

At $\sigma=0,\pi$, the vector fields $\mathcal{O}_{n}=\pm 2in\tau\partial_{\tau}$ do not contain any nonzero $\partial_{\sigma}$ terms and thus satisfy the boundary conditions.

In terms of vector fields on the cylinder, the algebra $\mathcal{O}(-1)$ preserves a different choice of boundaries, in this case at $\sigma=\frac{\pi}{2},\frac{3\pi}{2}$. This can be seen as follows:

[TABLE]

Therefore, at $\sigma=\frac{\pi}{2},\frac{3\pi}{2}$ we have $\mathcal{O}_{2n}^{(-1)}=\pm 4in\tau\partial_{\tau}$ and $\mathcal{O}_{2n+1}^{(-1)}=\pm(4n+2)\tau\partial_{\tau}$, which satisfy the boundary conditions.

When working over $\mathbb{C}$, this can be undone by a simple change of coordinates to $\sigma^{\prime}=\sigma-\frac{\pi}{2}$, with the resulting boundary conditions in terms of $\sigma^{\prime}$ being equivalent to the original one. Consequently, one would expect to have recovered $\mathcal{O}$. Indeed, this change of coordinates is just a rescaling of $\mathcal{O}$ by $i$, which reiterates the fact that the authors of the original paper were working over $\mathbb{C}$. However, Corollary 4.23 implies that the boundary conditions at $\sigma=\frac{\pi}{2},\frac{3\pi}{2}$ result in a non-isomorphic Lie algebra when working over $\mathbb{R}$.

We now focus on the case of $\mathcal{O}=\mathcal{O}(1)$.

Notation 4.25**.**

We make the following definitions:

[TABLE]

where $n\in\mathbb{Z}_{+}$.

Of course, $\{\mathcal{O}_{n}\mid n\geq 1\}$ and $\{u_{n}\mid n\geq 1\}$ are two bases for the same Lie algebra $\mathcal{O}$, so it is natural to ask how they are related. The following result answers this.

Lemma 4.26**.**

We have

[TABLE]

for all $n\geq 1$.

Proof.

First, note that $u_{1}=-(t^{2}-1)\partial=L_{1}-L_{-1}=\mathcal{O}_{1}$. Furthermore,

[TABLE]

We now compute for $n\geq 2$:

[TABLE]

Pairing up $t^{m}$ with $t^{-m}$ and using (4.2), we get

[TABLE]

Applying the identity

[TABLE]

the result now follows. ∎

4.3. Change of basis for $\mathcal{P}_{b}$

In Theorem 4.16, it was shown that the change of variables $s\mapsto t$ yields the isomorphism $\mathcal{O}\cong\mathfrak{f}=\Bbbk[t](t^{2}-4)\partial$. Therefore, it is natural to ask if this change of variables gives a similar description of $\mathcal{P}$. In this section, we show that this is indeed the case: the change of variables will allow us to define a new basis for $\mathcal{P}$. We will work in a slightly more general setting: we consider the change of variables on the entire family of $\mathcal{O}$-modules $\mathcal{P}_{b}$ for arbitrary $b\in\Bbbk$ (see Definition 2.6). Just like for $\mathcal{O}$, it will be useful to have a generating set for the $\mathcal{O}$-module $\mathcal{P}_{b}$.

Lemma 4.27**.**

For all $b\in\Bbbk^{*}$, the $\mathcal{O}$-module $\mathcal{P}_{b}$ is generated by $P^{(b)}_{0}$. For $b=0$, the $\mathcal{O}$-module $\mathcal{P}_{0}$ is generated by $P_{0}^{(0)}$ and $P_{1}^{(0)}$.

Proof.

If $b\neq 0$, this follows immediately (2.4), since

[TABLE]

for all $n\geq 1$. If $b=0$, then (2.4) gives

[TABLE]

for all $n\in\mathbb{N}$. The result now follows by an easy induction. ∎

To describe how the change of variables $s\mapsto t$ affects $\mathcal{P}_{b}$, we first introduce some notation. As above, we make the identification $\partial_{s}=\frac{1}{s^{\prime}}\partial$, so that $\partial_{s}(s)=1$. Letting $r\coloneqq t-t^{-1}$, we have

[TABLE]

Note that

[TABLE]

In other words, $r$ is a square root of $s^{2}-4$.

Recall that $\mathcal{P}_{b}$ is the $\mathcal{O}$-submodule of $I(0,b)$ spanned by $P^{(b)}_{n}=I_{n}+I_{-n}$. As mentioned in Definition 2.2, we view $I(0,b)$ as $I(0,b)=t^{-b}\Bbbk[t,t^{-1}]dt^{b}$ with $I_{n}=t^{n-b}\ dt^{b}$. With this perspective, we have

[TABLE]

Now, we want to express $\mathcal{P}_{b}$ in terms of polynomials in $s$, just like we did with $\mathcal{O}$. By definition of the Kähler differential, we have $ds=s^{\prime}\ dt$. It follows that $ds^{b}=(s^{\prime}\ dt)^{b}=(s^{\prime})^{b}\ dt^{b}$, and therefore

[TABLE]

so $t^{-b}\ dt^{b}=r^{-b}\ ds^{b}$. The next result shows what happens to the generators of $\mathcal{P}_{b}$ when written in terms of $ds^{b}$.

Lemma 4.28**.**

For $b\in\Bbbk$, we have $P^{(b)}_{0}=-2r^{-b}\ ds^{b}$ and $P^{(b)}_{1}=-sr^{-b}\ ds^{b}$.

Proof.

Follows immediately from (4.5) and (4.6) upon recalling that $s=t+t^{-1}$. ∎

Inspired by Lemma 4.28, we now introduce the following notation.

Notation 4.29**.**

For $b\in\Bbbk$ and $n\in\mathbb{N}$, we define $v_{n}^{(b)}\coloneqq-s^{n}r^{-b}\ ds^{b}=-(t+t^{-1})^{n}t^{-b}\ dt^{b}\in I(0,b)$.

By Lemma 4.28, we have $v_{0}^{(b)}=\frac{1}{2}P^{(b)}_{0}$ and $v_{1}^{(b)}=P^{(b)}_{1}$. In fact, as we show next, $\{v_{n}^{(b)}\mid n\in\mathbb{N}\}$ is another basis for $\mathcal{P}_{b}$.

Theorem 4.30**.**

For all $b\in\Bbbk$, we have

[TABLE]

The $\mathcal{O}$-action on $\mathcal{P}_{b}$ with this basis is given by

[TABLE]

for all $n\geq 1$ and $m\geq 0$. In basis-free notation, the $\mathcal{O}$-action is

[TABLE]

for all $g,h\in\Bbbk[s]$.

Proof.

We start by computing $u_{n}\cdot v_{m}^{(b)}$. To that end, note that

[TABLE]

and thus

[TABLE]

where we used that $r^{2}=s^{2}-4$ from (4.4) in the last equality. Therefore, we have

[TABLE]

as required. It follows that

[TABLE]

Thus, letting $V\coloneqq\operatorname{span}\{v_{n}^{(b)}\mid n\in\mathbb{N}\}=\Bbbk[s]r^{-b}\ ds^{b}$, we see that $V$ is an $\mathcal{O}$-module. Since $P^{(b)}_{0}=2v_{0}^{(b)}$ and $P^{(b)}_{1}=v_{1}^{(b)}$ are elements of $V$, Lemma 4.27 implies that $\mathcal{P}_{b}\subseteq V$.

For the other inclusion $V\subseteq\mathcal{P}_{b}$, it suffices to show that $v_{0}^{(b)}$ and $v_{1}^{(b)}$ generate $V$. We have

[TABLE]

for all $n$. The result now follows by an easy induction. ∎

The next result, which shows that $\mathcal{P}_{b}$ is a reducible $\mathcal{O}$-module for all $b\in\Bbbk$, follows immediately from Theorem 4.30.

Corollary 4.31**.**

Let $b\in\Bbbk$. Then $(s+2)\mathcal{P}_{b}$ and $(s-2)\mathcal{P}_{b}$ are proper $\mathcal{O}$-submodules of $\mathcal{P}_{b}$. Consequently, $\mathcal{P}_{b}$ is a reducible $\mathcal{O}$-module for all $b\in\Bbbk$.

Proof.

If $h$ is a multiple of $s\pm 2$ in (4.7), then $[(s^{2}-4)g\partial_{s},hr\ ds^{-1}]$ is also a multiple of $s\pm 2$ for all $g\in\Bbbk[s]$, so we see that $(s\pm 2)\mathcal{P}_{b}$ is an $\mathcal{O}$-submodule of $\mathcal{P}_{b}$. ∎

On the other hand, $\mathcal{P}_{b}$ is indecomposable as an $\mathcal{O}$-module, provided $b\neq 0$.

Lemma 4.32**.**

If $b\in\Bbbk\setminus\{0\}$, then $\mathcal{P}_{b}$ is an indecomposable $\mathcal{O}$-module. Furthermore, $\mathcal{P}_{0}$ decomposes as

[TABLE]

Proof.

The first part of the result follows from the following easy claim: if $M$ is a nonzero submodule of $\mathcal{P}_{b}$ with $b\neq 0$, then $M$ has finite codimension in $\mathcal{P}_{b}$. The second part is immediate from (2.4) and Corollary 4.31. ∎

We finish by considering some consequences of Theorem 4.30 for $\mathcal{P}=\mathcal{P}_{-1}$. First, we simplify the notation for the new basis of $\mathcal{P}$, since this is the main $\mathcal{O}$-module of interest in this paper.

Notation 4.33**.**

We define $v_{n}\coloneqq v_{n}^{(-1)}=-s^{n}r\ ds^{-1}=-(t+t^{-1})^{n}t\ dt^{-1}\in\mathcal{P}$ for $n\in\mathbb{N}$.

The next result computes the bracket of the centreless BCCA $\mathfrak{b}$ with the new bases for $\mathcal{O}$ and $\mathcal{P}$.

Corollary 4.34**.**

We have $\mathfrak{b}=\operatorname{span}\{u_{n},v_{m}\mid n\geq 1,m\in\mathbb{N}\}$. The bracket of $\mathfrak{b}$ with this basis is given by

[TABLE]

Proof.

Follows immediately from Theorems 4.16 and 4.30. ∎

As was done in Lemma 4.26, we consider the relationship between the two bases $P_{n}$ and $v_{n}$ of $\mathcal{P}$. Since $P_{0}=2v_{0}$, we need to be a bit careful, but otherwise the proof is nearly identical to that of Lemma 4.26.

Lemma 4.35**.**

For all $n\in\mathbb{N}$, we have

[TABLE]

where $Q_{n}\coloneqq\begin{cases}P_{n},&\text{if }n\neq 0,\\ \frac{1}{2}P_{0},&\text{if }n=0.\end{cases}$

Proof.

Similar to Lemma 4.26, except we apply the identity

[TABLE]

in the last part of the proof. ∎

Thanks to Lemmas 4.26 and 4.35, we can easily deduce what the central extension of $\mathfrak{b}$ giving rise to $\widehat{\mathfrak{b}}$ looks like using the new basis $\{u_{n},v_{m}\mid n\geq 1,m\geq 0\}$ of $\mathfrak{b}$. As the next result shows, the central extension becomes rather unwieldy with this choice of basis.

Corollary 4.36**.**

View $u_{n}$ and $v_{n}$ as elements of the (centrally extended) BCCA $\widehat{\mathfrak{b}}$. Then

[TABLE]

for all $n,m$. ∎

Remark 4.37**.**

The centreless BCCA is strongly almost-graded according to [Sch17, Definition 4.1]: setting $\mathfrak{b}_{n}\coloneqq\operatorname{span}\{u_{n},v_{n}\}$ for $n\in\mathbb{N}$ (where $u_{0}=0$), we see that

[TABLE]

for all $n,m\in\mathbb{N}$, by Corollary 4.34. However, Corollary 4.36 shows that cocycle on $\mathfrak{b}$ giving rise to $\widehat{\mathfrak{b}}$ is not compatible with the almost-grading, in the sense that is impossible to extend the almost-grading of $\mathfrak{b}$ to an almost-grading of $\widehat{\mathfrak{b}}$. This motivates our approach in later sections, where we focus on the centreless BCCA, since we would like to use the new basis $\{u_{n},v_{m}\mid n\geq 1,m\geq 0\}$ to study the representation theory of the BCCA. This becomes very difficult if we have to deal with the central extension as described in Corollary 4.36.

Furthermore, representations of the BCCA where $C_{M}$ acts trivially are to be expected when one takes the tensionless limit on the spectrum of the tensile open bosonic string. This is because of how the tensionless limit of the open bosonic string corresponds to a Lie algebra contraction of its worldsheet symmetry algebra (which is simply the Virasoro algebra). If one were to define this contraction [BCC*+*25b, Equation (8)] at the level of representations, then one would obtain $\widehat{\mathfrak{b}}$-modules on which $C_{M}$ acts trivially. Thus, we note that the study of centreless BCCA modules is also physically motivated.

Finally, we consider what happens to $\mathcal{P}$ under the change of variables $s\mapsto t$. Since $r$ is a square root of $s^{2}-4$ by (4.4), we get that $r\mapsto\sqrt{t^{2}-4}=\sqrt{f}$ under this change of variables.

Corollary 4.38**.**

Under the change of variables $s\mapsto t$, the $\mathcal{O}$-module $\mathcal{P}$ corresponds to the $\mathfrak{f}$-module $\mathfrak{m}\coloneqq\Bbbk[t]\sqrt{f}\ dt^{-1}$. In other words, $\mathfrak{b}\cong\mathfrak{f}\ltimes\mathfrak{m}$.

Proof.

Follows from Theorems 4.16 and 4.30, and the observation that $r=\sqrt{s^{2}-4}$ from (4.4). ∎

We finish by summarising the results of this section for the centreless BCCA.

• •

We have defined two different bases for the centreless BCCA, denoted by $\{\mathcal{O}_{n},\mathcal{P}_{m}\mid n\geq 1,m\geq 0\}$ and $\{u_{n},v_{m}\mid n\geq 1,m\geq 0\}$. The Lie bracket of $\mathfrak{b}$ with these two bases is given by

[TABLE]

[TABLE]

• •

In basis-free notation, we have $\mathcal{O}=\Bbbk[t+t^{-1}](t^{2}-1)\partial$ and $\mathcal{P}=\Bbbk[t+t^{-1}]t\ dt^{-1}$. Letting $s=t+t^{-1}$, $r=t-t^{-1}$, and $\partial_{s}=\frac{1}{s^{\prime}}\partial=\frac{t^{2}}{t^{2}-1}\partial$, we can also write $\mathcal{O}$ and $\mathcal{P}$ as

[TABLE]

• •

There is an isomorphism $\Phi\colon\mathcal{O}\ltimes\mathcal{P}\to\mathfrak{f}\ltimes\mathfrak{m}$ given by the change of variables $s\mapsto t$, where $\mathfrak{f}=f\mathbb{W}_{1}=(t^{2}-4)\mathbb{W}_{1}$ and $\mathfrak{m}=\Bbbk[t]\sqrt{f}\ dt^{-1}$. Explicitly, $\Phi(u_{n})=f_{n}$ and $\Phi(v_{n})=-t^{n}\sqrt{f}\ dt^{-1}$ for all $n$, where $f_{n}=L_{n}-4L_{n-2}$. In basis-free notation,

[TABLE]

for all $p\in\Bbbk[t]$. In particular, $\mathcal{O}$ is isomorphic to a subalgebra of $\mathbb{W}_{1}$ of finite codimension.

5. Filtering the BCCA

5.1. Defining the filtration

Using the alternate basis for the BCCA from Corollary 4.34, we construct a descending filtration for $\mathfrak{b}$, thus allowing us to define and study Whittaker modules for the BCCA. We first filter the subalgebra $\mathcal{O}$ of the BCCA. It is convenient to introduce a shift in the basis $u_{n}$ of $\mathcal{O}$: define

[TABLE]

for $n\geq-1$. We can now construct a filtration of $\mathcal{O}$.

Lemma 5.1**.**

For $n\in\mathbb{Z}$, define

[TABLE]

Then $\mathcal{F}$ is a filtration of $\mathcal{O}$.

Proof.

We have

[TABLE]

so we see that $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{m}\mathcal{O}]\subseteq\mathcal{F}_{n+m}\mathcal{O}$ for all $n,m$. ∎

We now extend the filtration $\mathcal{F}$ to the full BCCA. As before, we must shift the basis $v_{n}$: define

[TABLE]

for $n\geq-1$.

Proposition 5.2**.**

For $n\in\mathbb{Z}$, define

[TABLE]

Then $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{m}\mathcal{P}]\subseteq\mathcal{F}_{n+m}\mathcal{P}$ for all $n,m\in\mathbb{Z}$. Consequently, we can filter the full centreless BCCA as follows:

[TABLE]

Proof.

For all $n,m\geq-1$, we have

[TABLE]

and thus $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{m}\mathcal{P}]\subseteq\mathcal{F}_{n+m}\mathcal{P}$. It is now clear that $\mathcal{F}$ is a filtration of the entirety of $\mathfrak{b}$, by Lemma 5.1. In other words, $[\mathcal{F}_{n}\mathfrak{b},\mathcal{F}_{m}\mathfrak{b}]\subseteq\mathcal{F}_{n+m}\mathfrak{b}$, and

[TABLE]

We finish this section by considering the associated graded algebra to the filtration $\mathcal{F}$.

Lemma 5.3**.**

The associated graded Lie algebras to the filtration $\mathcal{F}$ are

1. (1)

$\operatorname{gr}_{\mathcal{F}}\mathcal{O}\cong\mathbb{W}_{1}$, and 2. (2)

$\operatorname{gr}_{\mathcal{F}}\mathfrak{b}\cong\mathbb{W}_{1}\ltimes\operatorname{ad}(\mathbb{W}_{1})$, where $\operatorname{ad}(\mathbb{W}_{1})$ is another copy of $\mathbb{W}_{1}$, viewed as the adjoint module of $\mathbb{W}_{1}$ with abelian Lie bracket.

Proof.

We have

[TABLE]

Thus, the Lie algebra homomorphism $\mathcal{U}_{n}+\mathcal{F}_{n+1}\mathcal{O}\mapsto-4L_{n}$ establishes the isomorphism $\operatorname{gr}_{\mathcal{F}}\mathcal{O}\cong\mathbb{W}_{1}$. Similarly, we have

[TABLE]

It is now clear that $\operatorname{gr}_{\mathcal{F}}\mathfrak{b}\cong\mathbb{W}_{1}\ltimes\operatorname{ad}(\mathbb{W}_{1})$ using a similar isomorphism to the above, as required. ∎

5.2. The derived subalgebras of the filtered subspaces

To study Whittaker functions on $\mathcal{O}$ and $\mathfrak{b}$, we must study their Whittaker functions. By definition, these are Lie algebra homomorphisms $\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ or $\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$, for some $n\in\mathbb{N}$. Since $\Bbbk$ is an abelian Lie algebra, these homomorphisms must necessarily vanish on the derived subalgebras $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$ or $[\mathcal{F}_{n}\mathfrak{b},\mathcal{F}_{n}\mathfrak{b}]$. We compute these in this section.

First, note that $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]\subseteq\mathcal{F}_{2n+1}\mathcal{O}$, by (5.1). We now prove that $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$ is spanned by $\mathcal{U}_{2n+2+k}-4\,\mathcal{U}_{2n+k}$ for $k\geq 1$.

Lemma 5.4**.**

For all $n\geq-1$, the derived subalgebra $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$ has the following form:

[TABLE]

Consequently, the set $\{\mathcal{U}_{2n+3+k}-4\,\mathcal{U}_{2n+1+k}\mid k\in\mathbb{N}\}$ is a basis for $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$.

Proof.

Recall from Theorem 4.16 that $\mathcal{O}=(s^{2}-4)\Bbbk[s]\partial_{s}$, and that $\mathcal{F}_{n}\mathcal{O}=s^{n+1}\mathcal{O}$ by definition. Therefore,

[TABLE]

by (4.1). Simplifying, we get

[TABLE]

Noting that

[TABLE]

it is now clear that the result follows. ∎

We now proceed as in Lemma 5.4 to compute the derived subalgebra $[\mathcal{F}_{n}\mathfrak{b},\mathcal{F}_{n}\mathfrak{b}]$ for the filtration of the full BCCA. Similarly to the above, (5.2) implies that $\mathcal{F}_{n}\mathcal{O}\cdot\mathcal{F}_{n}\mathcal{P}\subseteq\mathcal{F}_{2n+1}\mathcal{P}$. In fact, this is an equality: we have $\mathcal{F}_{n}\mathcal{O}\cdot\mathcal{F}_{n}\mathcal{P}=\mathcal{F}_{2n+1}\mathcal{P}$, as we prove next.

Lemma 5.5**.**

For all $n\geq-1$, we have $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{P}]=\mathcal{F}_{2n+1}\mathcal{P}$. Consequently,

[TABLE]

and therefore the set $\{\mathcal{U}_{2n+3+k}-4\,\mathcal{U}_{2n+1+k},\mathcal{V}_{2n+1+k}\mid k\in\mathbb{N}\}$ is a basis for $[\mathcal{F}_{n}\mathfrak{b},\mathcal{F}_{n}\mathfrak{b}]$.

Proof.

For all $m\geq n$, we have

[TABLE]

by (5.2). The result follows. ∎

5.3. The abelianisations of the filtered subspaces

We finish this section by computing the dimensions of the abelianisations of $\mathcal{F}_{n}\mathcal{O}$ and $\mathcal{F}_{n}\mathfrak{b}$. This computation will tell us exactly how many degrees of freedom one has when defining a Whittaker function on $\mathcal{O}$ or on $\mathfrak{b}$. The following easy observation will be useful for this purpose.

Lemma 5.6**.**

For any $g\in\Bbbk[t]$, the codimension of $g\mathbb{W}_{1}$ in $\mathbb{W}_{1}$ is $\deg(g)$. ∎

Of course, Lemma 5.6 also implies that $g(s)\Bbbk[s]\partial_{s}$ has codimension $\deg(g)$ in $\Bbbk[s]\partial_{s}$, via the change of variables $s\mapsto t$.

Proposition 5.7**.**

We have

[TABLE]

for all $n\geq-1$.

Proof.

By Theorem 4.16, we have

[TABLE]

This is a subalgebra of $\Bbbk[s]\partial_{s}$ of codimension $n+3$, by Lemma 5.6. On the other hand, Lemma 5.6 also implies that

[TABLE]

is a subalgebra of $\Bbbk[s]\partial_{s}$ of codimension $2n+6$. Therefore, $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$ has codimension $n+3$ in $\mathcal{F}_{n}\mathcal{O}$, as required.

For the second part of the result, we observe that the codimension of $\mathcal{F}_{m}\mathcal{P}$ in $\mathcal{F}_{n}\mathcal{P}$ is $m-n$, for all $m\geq n\geq-1$. Now, Lemmas 5.5 and 5.6 imply that

[TABLE]

which concludes the proof. ∎

More precisely, write $\overline{x}$ for the image of $x$ in the abelianisation $\mathcal{F}_{n}\mathfrak{b}/[\mathcal{F}_{n}\mathfrak{b},\mathcal{F}_{n}\mathfrak{b}]$ or $\mathcal{F}_{n}\mathcal{O}/[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$, where $x\in\mathcal{F}_{n}\mathfrak{b}$ or $x\in\mathcal{F}_{n}\mathcal{O}$, and $n\in\mathbb{Z}$. Then Proposition 5.7 implies that $\{\overline{\mathcal{U}}_{n},\dots,\overline{\mathcal{U}}_{2n+2}\}$ forms a basis of $\mathcal{F}_{n}\mathcal{O}/[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$, and $\{\overline{\mathcal{U}}_{n},\dots,\overline{\mathcal{U}}_{2n+2}\}\cup\{\overline{\mathcal{V}}_{n},\dots,\overline{\mathcal{V}}_{2n}\}$ forms a basis of $\mathcal{F}_{n}\mathfrak{b}/[\mathcal{F}_{n}\mathfrak{b},\mathcal{F}_{n}\mathfrak{b}]$.

The simplest application of Proposition 5.7 is that we can completely classify all one-dimensional representations of $\mathcal{O}$ and $\mathfrak{b}$.

Definition 5.8**.**

Let $\lambda,\mu\in\Bbbk$, and define the $\mathcal{O}$-module $\Bbbk_{\lambda,\mu}$ as follows: as a vector space, $\Bbbk_{\lambda,\mu}=\Bbbk 1_{\lambda,\mu}$, and the $\mathcal{O}$-action is given by

[TABLE]

The action of the other basis elements is determined by these and the formula

[TABLE]

For example, we have $\mathcal{U}_{1}\cdot 1_{\lambda,\mu}=4\lambda 1_{\lambda,\mu}$ and $\mathcal{U}_{2}\cdot 1_{\lambda,\mu}=4\mu 1_{\lambda,\mu}$. In fact, all one-dimensional representations of $\mathcal{O}$ and $\mathfrak{b}$ are of the above form, which is an immediate consequence of Lemmas 5.4 and 5.5, and Proposition 5.7.

Corollary 5.9**.**

If $M$ is a one-dimensional $\mathcal{O}$-representation, then $M\cong\Bbbk_{\lambda,\mu}$ for some $\lambda,\mu\in\Bbbk$. If $M$ is a one-dimensional $\mathfrak{b}$-representation, then $\mathcal{P}\cdot M=0$, so $M$ can be viewed as a representation over $\mathfrak{b}/\mathcal{P}\cong\mathcal{O}$. ∎

6. Whittaker modules

Having defined and analysed the filtrations for $\mathcal{O}$ and $\mathfrak{b}$, we now study Whittaker modules for these Lie algebras (recall Lemma 2.11).

6.1. Whittaker modules for $\mathcal{O}$

We start by defining Whittaker functions for $\mathcal{O}$, which in turn define universal Whittaker modules for $\mathcal{O}$. We further characterise precisely when these modules are irreducible.

Definition 6.1**.**

A Whittaker function on $\mathcal{O}$ is a Lie algebra homomorphism $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ for some $n\in\mathbb{N}$.

As we show next, a Whittaker function $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ is determined by its values on $\mathcal{U}_{n},\dots,\mathcal{U}_{2n+2}$.

Lemma 6.2**.**

A Whittaker function $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ is determined by its values on $\mathcal{U}_{n},\dots,\mathcal{U}_{2n+2}$, and $\varphi_{n}(\mathcal{U}_{m+2})=4\varphi_{n}(\mathcal{U}_{m})$ for all $m\geq 2n+1$.

Proof.

Follows immediately from Lemma 5.4 and Proposition 5.7. ∎

Definition 6.3**.**

Let $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ be a Whittaker function and $\Bbbk_{\varphi_{n}}=\Bbbk 1_{\varphi_{n}}$ be the one-dimensional $\mathcal{F}_{n}\mathcal{O}$-module defined by $\varphi_{n}$, in other words,

[TABLE]

for all $x\in\mathcal{F}_{n}\mathfrak{b}$. Define the universal Whittaker module for $\mathcal{O}$ of type $\varphi_{n}$ to be the induced module

[TABLE]

Remark 6.4**.**

Note that $\Bbbk_{\varphi_{n}}$ is a continuous module for $\mathcal{F}_{n}\mathcal{O}$ in the sense of Rudakov [Rud74] if and only if $\varphi_{n}(\mathcal{U}_{2n+1})=\varphi_{n}(\mathcal{U}_{2n+2})=0$. In fact, if $\varphi_{n}(\mathcal{U}_{2n+1})\neq 0$, then $\varphi(\mathcal{U}_{2n+2k+1})\neq 0$ for all $k\geq 0$, and if $\varphi_{n}(\mathcal{U}_{2n+2})\neq 0$, then $\varphi_{n}(\mathcal{U}_{2m+2k+2})\neq 0$ for all $k\geq 0$, by Lemma 6.2.

Remark 6.5**.**

The Lie algebra $\mathcal{F}_{0}\mathcal{O}$ is not quasi-nilpotent [Buz24, Remark 3.5], so $M_{\varphi_{0}}$ is not a Whittaker module according to Definition 2.9. As we will see, the criteria for irreducibility for $M_{\varphi_{0}}$ and their proofs are slightly different to those for $M_{\varphi_{n}}$ with $n\geq 1$, highlighting the importance of the quasi-nilpotence condition. However, to avoid piling onto the nomenclature, we abuse terminology and opt to call $M_{\varphi_{0}}$ a universal Whittaker module.

We also highlight that $\mathcal{F}_{n}\mathcal{O}$ is not an ideal of $\mathcal{O}$ if $n\geq 0$. Therefore, the universal Whittaker modules $M_{\varphi_{n}}$ are never quasi-Whittaker modules according to Definition 2.12.

The following result is the main theorem of this subsection.

Theorem 6.6**.**

Let $n\in\mathbb{N}$, and let $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ be a Whittaker function. If $n=0$, the universal Whittaker module $M_{\varphi_{0}}$ is irreducible if and only if $\varphi_{0}(\mathcal{U}_{2})\neq 4\varphi_{0}(\mathcal{U}_{0})$. If $n\geq 1$, the universal Whittaker module $M_{\varphi_{n}}$ is irreducible if and only if $\varphi_{n}(\mathcal{U}_{2n+1})\neq 4\varphi_{n}(\mathcal{U}_{2n-1})$ or $\varphi_{n}(\mathcal{U}_{2n+2})\neq 4\varphi_{n}(\mathcal{U}_{2n})$.

We now proceed to prove that $M_{\varphi_{n}}$ is reducible if the condition from Theorem 6.6 is not satisfied, proving one direction of Theorem 6.6. As mentioned in Remark 6.5, the proofs are slightly different depending on whether $n=0$ or $n\geq 1$.

Lemma 6.7**.**

Let $\varphi_{0}\colon\mathcal{F}_{0}\mathcal{O}\to\Bbbk$ be a Whittaker function with $\varphi_{0}(\mathcal{U}_{2})=4\varphi_{0}(\mathcal{U}_{0})$. Then $U(\mathcal{O})\mathcal{U}_{-1}\cdot 1_{\varphi_{0}}$ is a proper submodule of the universal Whittaker module $M_{\varphi_{0}}$.

Proof.

Let $\lambda_{i}\coloneqq\varphi_{n}(\mathcal{U}_{i})$ for $i\geq 0$, and $w\coloneqq(\lambda_{1}-4\,\mathcal{U}_{-1})\cdot 1_{\varphi_{0}}=[\mathcal{U}_{0},\mathcal{U}_{-1}]\cdot 1_{\varphi_{0}}$. We claim that $U(\mathcal{O})\cdot w$ is a proper $\mathcal{O}$-submodule of $M_{\varphi_{0}}$.

Lemma 6.2 together with the assumption that $\lambda_{2}=4\lambda_{0}$ imply that $\lambda_{n+2}=4\lambda_{n}$ for all $n\geq 0$. Then

[TABLE]

Now letting $n\geq 1$, we have

[TABLE]

where we used that $\lambda_{n+1}=4\lambda_{n-1}$ in the last equality, since $n\geq 1$. This proves the claim, and the result follows. ∎

Remark 6.8**.**

When $\lambda_{2}=4\lambda_{0}$, the submodule $U(\mathcal{O})(\lambda_{1}-4\,\mathcal{U}_{-1})\cdot 1_{\varphi_{0}}$ in the proof of Lemma 6.7 is isomorphic to $M_{\varphi_{0}^{\prime}}$, where $\varphi^{\prime}_{0}\colon\mathcal{F}_{0}\mathcal{O}\to\Bbbk$ is the Whittaker function with $\varphi^{\prime}_{0}(\mathcal{U}_{0})=\lambda_{0}-4$ and $\varphi^{\prime}_{0}(\mathcal{U}_{n})=\lambda_{n}$ for $n\geq 1$.

In this case, the quotient module $M_{\varphi_{0}}/(U(\mathcal{O})(\lambda_{1}-4\,\mathcal{U}_{-1})\cdot 1_{\varphi_{0}})$ is a $1$-dimensional $\mathcal{O}$-module determined by $\mathcal{U}_{-1}\cdot 1_{\varphi_{0}}=\frac{\lambda_{1}}{4}1_{\varphi_{0}},\mathcal{U}_{0}\cdot 1_{\varphi_{0}}=\lambda_{0}1_{\varphi_{0}}$.

The case $n\geq 1$ follows from the existence of an analogous submodule.

Lemma 6.9**.**

Let $n\geq 1$ and let $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ be a Whittaker function with $\varphi_{n}(\mathcal{U}_{2n+1})=4\varphi_{n}(\mathcal{U}_{2n-1})$ and $\varphi_{n}(\mathcal{U}_{2n+2})=4\varphi_{n}(\mathcal{U}_{2n})$. Then $U(\mathcal{O})\mathcal{U}_{n-1}\cdot 1_{\varphi_{n}}$ is a submodule of the universal Whittaker module $M_{\varphi_{n}}$.

Proof.

Let $\lambda_{i}\coloneqq\varphi_{n}(\mathcal{U}_{i})$ for $i\geq n$, and $w\coloneqq\mathcal{U}_{n-1}\cdot 1_{\varphi_{n}}$. We claim that $U(\mathcal{O})\cdot w$ is a proper submodule of $M_{\varphi_{n}}$.

Lemma 6.2 together with the assumptions that $\lambda_{2n+1}=4\lambda_{2n-1}$ and $\lambda_{2n+2}=4\lambda_{2n}$ imply that $\lambda_{k+2}=4\lambda_{k}$ for all $k\geq 2n-1$. For $m\geq n$, we have

[TABLE]

since $m+n-1\geq 2n-1$, where we used that $\lambda_{k+2}=4\lambda_{k}$ for all $k\geq 2n-1$.

The above shows that $\Bbbk w$ is a one-dimensional $\mathcal{F}_{n}\mathcal{O}$-module isomorphic to $\Bbbk_{\varphi_{n}}$. It is now easy to see that $U(\mathcal{O})\cdot w$ is a proper submodule of $M_{\varphi_{n}}$ isomorphic to $M_{\varphi_{n}}$ itself. ∎

Note that together Lemmas 6.7 and 6.9 prove one direction of Theorem 6.6. To prove the converse, we use the theory of local functions and the orbit method, which we introduce in the next subsection.

6.2. Local functions and the orbit method for $\mathcal{O}$

Kirillov’s orbit method [Kir04] is a guiding principle in Lie theory, establishing a correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits. In this subsection, we explain the connection between the Whittaker functions on $\mathcal{O}$ and the orbit method for the Lie algebra $\mathbb{W}_{1}$ constructed in [Pha25]. This will allow us to use results in [Pha25] to prove the other direction of Theorem 6.6. Note that in this subsection, we view $\mathbb{W}_{1}=\operatorname{Der}(\Bbbk[s])=\Bbbk[s]\partial_{s}$ (in other words, we use the variable $s$ instead of $t$), so that we can view $\mathcal{O}$ as the subalgebra $(s^{2}-4)\mathbb{W}_{1}$ of $\mathbb{W}_{1}$.

We start by recalling the notion of a local function on $\mathbb{W}_{1}$, originally defined in [PS23].

Definition 6.10**.**

Letting $x,\alpha_{0},\alpha_{1},\dots,\alpha_{n}\in\Bbbk$ with $\alpha_{n}\neq 0$, we define a one-point local function $\chi_{x;\alpha_{0},\dots,\alpha_{n}}\in\mathbb{W}_{1}^{\ast}$ by

[TABLE]

where $f\in\Bbbk[s]$ and $f^{(i)}$ denotes taking the $i^{\text{th}}$ derivative of $f$ with respect to $s$. We call $x$ the base point of $\chi$ and $n$ the order of $\chi$.

A local function $\chi$ on $\mathbb{W}_{1}$ is a sum of finitely many functions of the form  (6.1) with (possibly) distinct $x$. In other words,

[TABLE]

where $\chi_{i}$ is a one-point local function based at $x_{i}\in\Bbbk$ of order $n_{i}$, and the $x_{i}$ are pairwise distinct.

Next, we recall the notion of a polarisation for a local function on $\mathbb{W}_{1}$ in [Pha25].

Definition 6.11**.**

For every $\chi\in\mathbb{W}_{1}^{*}$, we define a bilinear form

[TABLE]

A polarisation $\mathfrak{p}_{\chi}$ of $\chi$ is a finite-codimensional subalgebra of $\mathbb{W}_{1}$ that is maximally totally isotropic subspace with respect to $B_{\chi}$, in other words, $B_{\chi}([\mathfrak{p}_{\chi},\mathfrak{p}_{\chi}])=0$.

Remark 6.12**.**

In general, an arbitrary element $\chi\in\mathfrak{g}^{*}$ for a Lie algebra $\mathfrak{g}$ might not have a polarisation. However, if $\chi$ is a local function on $\mathbb{W}_{1}$, then $\chi$ always has a polarisation [Pha25, Proposition 3.2]. In this case, the polarisation can moreover be chosen as a submodule-subalgebra of $\mathbb{W}_{1}$.

We consider an example of a local function and its polarisation to illustrate the concept.

Example 6.13** ([PS23]).**

Let $\chi\coloneqq\chi_{x;\nu_{0},\nu_{1}}$ be a one-point local function on $\mathbb{W}_{1}$ where $x\in\Bbbk$ and $(\nu_{0},\nu_{1})\in\Bbbk^{2}\setminus\{(0,0)\}$. In other words,

[TABLE]

Consider the submodule-subalgebra $(s-x)\mathbb{W}_{1}$ of $\mathbb{W}_{1}$. Certainly, $(s-x)\mathbb{W}_{1}$ has codimension 1 in $\mathbb{W}_{1}$. Moreover, (4.1) implies that

[TABLE]

and therefore $(s-x)\mathbb{W}_{1}$ is a totally isotropic subspace of $\mathbb{W}_{1}$ with respect to $B_{\chi}$. Note that $(s-x)\mathbb{W}_{1}$ is maximally totally isotropic by a dimension count [Pha25, Lemma 3.1], and therefore is a polarisation of $\chi$.

Remark 6.14**.**

The polarisation $(s-x)\mathbb{W}_{1}$ is in fact the unique polarisation for the local function $\chi_{x;\nu_{0},\nu_{1}}$ (see, for example, [PS23, Section 7.1] for a proof). However, polarisations for local functions of $\mathbb{W}_{1}$ are in general not unique (for an example, see [Pha25, Example 3.3]).

Remark 6.15**.**

If $\mathfrak{q}$ is a subalgebra of a polarisation of a local function $\chi$, then $\chi([\mathfrak{q},\mathfrak{q}])=0$, and thus $\chi$ defines a one-dimensional representation of $\mathfrak{q}$:

[TABLE]

We now associate a local function to each Whittaker function $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$.

Proposition 6.16**.**

Let $n\in\mathbb{N}$ and let $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$ be a Whittaker function. Then there exists a local function $\chi=\chi_{0;\alpha_{0},\dots,\alpha_{2n+1}}+\chi_{2;\beta,\beta_{1}}+\chi_{-2;\gamma,\gamma_{1}}$ on $\mathbb{W}_{1}$ such that $\varphi_{n}={\left.\kern-1.2pt\chi\vphantom{\big|}\right|_{\mathcal{F}_{n}\mathcal{O}}}$.

Moreover, we can always choose $\alpha_{0},\beta,\gamma\in\Bbbk$ arbitrarily, and

[TABLE]

Proof.

Choose arbitrary $\alpha_{i},\beta,\beta_{1},\gamma,\gamma_{1}\in\Bbbk$, and let

[TABLE]

It is easy to check that $\chi$ vanishes on $s^{2n+2}(s^{2}-4)^{2}\mathbb{W}_{1}=[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$, thus $\mathcal{F}_{n}\mathcal{O}$ is contained in a polarisation of $\chi$. Hence, $\chi$ restricts to a one-dimensional representation ${\left.\kern-1.2pt\chi\vphantom{\big|}\right|_{\mathcal{F}_{n}\mathcal{O}}}$ of $\mathcal{F}_{n}\mathcal{O}$. Since both ${\left.\kern-1.2pt\chi\vphantom{\big|}\right|_{\mathcal{F}_{n}\mathcal{O}}}$ and $\varphi_{n}$ vanish on $[\mathcal{F}_{n}\mathcal{O},\mathcal{F}_{n}\mathcal{O}]$, we require the following system of equations to have ${\left.\kern-1.2pt\chi\vphantom{\big|}\right|_{\mathcal{F}_{n}\mathcal{O}}}=\varphi_{n}$:

[TABLE]

Note that (6.3) only exists for $n\geq 2$, and (6.4) only exists for $n\geq 1$. For all $n$, there are $n+3$ equations (from $\varphi_{n}(\mathcal{U}_{n})$ to $\varphi_{n}(\mathcal{U}_{2n+2})$) with $n+3$ variables (from $\alpha_{n+1}$ to $\alpha_{2n+1}$, as well as $\beta_{1}$, $\gamma_{1}$). Furthermore, there are no inconsistencies on the right hand side, thus there always exist $\alpha_{n+1},\dots,\alpha_{2n+1},\beta_{1},\gamma_{1}\in\Bbbk$ satisfying the above system. In other words, there always exists $\chi$ of the form (6.2) such that $\varphi_{n}={\left.\kern-1.2pt\chi\vphantom{\big|}\right|_{\mathcal{F}_{n}\mathcal{O}}}$. Since $\alpha_{0},\beta,\gamma$ are not involved in the system of equations, we can choose them arbitrarily.

The last part of the statement follows from the combination of (6.4) and (6.6) (if $n\geq 1$), and the combination of (6.5) and (6.7). ∎

For the rest of this section, we fix $n\in\mathbb{N}$ and a Whittaker function $\varphi_{n}\colon\mathcal{F}_{n}\mathcal{O}\to\Bbbk$. Given Proposition 6.16, we introduce notation for the local function associated to $\varphi_{n}$.

Notation 6.17**.**

For $k\geq n$, define $\lambda_{k}\coloneqq\varphi_{n}(\mathcal{U}_{k})$. Furthermore, using Proposition 6.16, choose $\alpha_{i},\beta_{1},\gamma_{1}\in\Bbbk$ such that $\varphi_{n}={\left.\kern-1.2pt\chi\vphantom{\big|}\right|_{\mathcal{F}_{n}\mathcal{O}}}$, where

[TABLE]

We will denote

[TABLE]

for $a,b\in\Bbbk$. For simplicity of notation, write $\theta\coloneqq\theta_{\beta_{1},\gamma_{1}}$, so that $\chi=\eta+\theta$.

Although it is not obvious at this point why we have introduced the family of local functions $\chi_{a,b}$, this soon become clear.

Remark 6.18**.**

The splitting $\chi=\eta+\theta$ was introduced because $\theta$ defines a Lie algebra homomorphism $\theta\colon\mathcal{O}\to\Bbbk$, since $\theta$ vanishes on $[\mathcal{O},\mathcal{O}]=(s^{2}-4)^{2}\mathbb{W}_{1}$. Therefore, $\Bbbk_{\theta}$ is a one-dimensional representation of $\mathcal{O}$. This is not true for $\eta$, and we will soon see that $\eta$ completely controls the reducibility of $M_{\varphi_{n}}$.

Having introduced the necessary background and notation, we proceed to prove the irreducibility of these representations under the conditions of Theorem 6.6, using the perspective of local functions. We begin this proof with realising the subalgebra $\mathcal{F}_{n}\mathcal{O}$ as a polarisation of the local function $\chi$, using results in [Pha25].

Lemma 6.19** ([Pha25, Proposition 3.2]).**

Let $\chi=\chi_{1}+\dots+\chi_{\ell}$ be a local function on $\mathbb{W}_{1}$, where $\chi_{i}$ is a one-point local function at $x_{i}$ with order $n_{i}$. Letting $m_{i}\coloneqq\left\lfloor\frac{n_{i}}{2}\right\rfloor$, the subalgebra

[TABLE]

is a polarisation of $\chi_{i}$. Furthermore,

[TABLE]

is a polarisation of $\chi$.

Corollary 6.20**.**

If $n=0$, then $\mathcal{F}_{0}\mathcal{O}$ is a polarisation of $\chi_{a,b}$ for all $a,b\in\Bbbk$. If $n\geq 1$, and $\lambda_{2n+1}\neq 4\lambda_{2n-1}$ or $\lambda_{2n+2}\neq 4\lambda_{2n}$, then $\mathcal{F}_{n}\mathcal{O}$ is a polarisation of $\chi_{a,b}$ for all $a,b\in\Bbbk$.

Proof.

Throughout this proof, we assume that $\lambda_{2n+1}\neq 4\lambda_{2n-1}$ or $\lambda_{2n+2}\neq 4\lambda_{2n}$ if $n\geq 1$.

We first claim that $s^{n+1}\mathbb{W}_{1}$ is a polarisation of $\eta$. If $n=0$, then by definition we have $\eta=\chi_{0;1,\alpha_{1}}$ for some $\alpha_{1}\in\Bbbk$. Therefore, $\mathfrak{p}_{\eta}=s\mathbb{W}_{1}$ is a polarisation of $\eta$, by Lemma 6.19. If $n\geq 1$, then Proposition 6.16 implies that either $\alpha_{2n}\neq 0$ or $\alpha_{2n+1}\neq 0$. Thus, by Lemma 6.19, it follows that $\mathfrak{p}_{\eta}=s^{n+1}\mathbb{W}_{1}$ is a polarisation of $\eta$. This proves the claim.

Recalling that $\theta_{a,b}=\chi_{2;1,a}+\chi_{-2;1,b}$ by definition, we also see that $\mathfrak{p}_{\theta_{a,b}}=(s^{2}-4)\mathbb{W}_{1}$ is a polarisation of $\theta_{a,b}$, by Lemma 6.19. We conclude that

[TABLE]

is a polarisation of $\chi_{a,b}$. ∎

The conditions under which $\operatorname{Ind}^{\mathbb{W}_{1}}_{\mathfrak{p}_{\chi}}\Bbbk_{\chi}$ is a simple $\mathbb{W}_{1}$-module are completely characterised in [Pha25]. We summarise the relevant results in the following theorem.

Theorem 6.21** ([Pha25, Corollary 4.27, Proposition 4.23]).**

Let $\chi=\chi_{1}+\dots+\chi_{\ell}$ be a local function on $\mathbb{W}_{1}$, where $\chi_{i}$ is a one-point local function at $x_{i}$ with order $n_{i}$. Letting $m_{i}\coloneqq\left\lfloor\frac{n_{i}}{2}\right\rfloor$, let

[TABLE]

be the polarisation of $\chi$ from Lemma 6.19. Then $\operatorname{Ind}_{\mathfrak{p}_{\chi}}^{\mathbb{W}_{1}}\Bbbk_{\chi}$ is an irreducible $\mathbb{W}_{1}$-module if and only if $n_{i}>0$ for all $i$.

Corollary 6.20 and Theorem 6.21 imply that $\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathbb{W}_{1}}\Bbbk_{\chi}$ can only be an irreducible $\mathbb{W}_{1}$-module if $\beta_{1}$ and $\gamma_{1}$ are nonzero. However, this is not guaranteed by Proposition 6.16. Thankfully, we can solve this issue: the reducibility of $M_{\varphi_{n}}=\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathcal{O}}\Bbbk_{\chi}$ does not depend on the values of $\beta_{1}$ and $\gamma_{1}$, as we aim to show next. In other words, the reducibility of $M_{\varphi_{n}}$ is equivalent to that of $\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathcal{O}}\Bbbk_{\chi_{a,b}}$ for any choice of $a,b\in\Bbbk$.

We now decompose the universal Whittaker module of $\mathcal{O}$ into the tensor product of an induced module of a local function and a one-dimensional representation. This helps us view the universal Whittaker module of $\mathcal{O}$ as a twisted induced module of the one-point local function $\eta$.

Lemma 6.22**.**

Let $a,b\in\Bbbk$, and let $\Bbbk_{\eta}$ and $\Bbbk_{\theta_{a,b}}$ be the one-dimensional representations of $\mathcal{F}_{n}\mathcal{O}$ and $\mathcal{O}$ induced by $\eta$ and $\theta_{a,b}$, respectively. Then

[TABLE]

In particular,

[TABLE]

Proof.

Note that $\Bbbk_{\chi_{a,b}}=\Bbbk_{\eta+\theta_{a,b}}\cong\Bbbk_{\eta}\otimes\Bbbk_{\theta_{a,b}}$ as a representation over $\mathcal{F}_{n}\mathcal{O}$. The proof now follows by [Dix96, Proposition 5.1.15] or [LZ20, Lemma 8]. The final sentence of the result follows by the observation that $\Bbbk_{\varphi_{n}}=\Bbbk_{\chi}$ as $\mathcal{F}_{n}\mathcal{O}$-modules. ∎

The reducibility of a module is preserved when taking the tensor product with a one-dimensional representation of a Lie algebra. Thus, by Lemma 6.22, the irreducibility of the universal Whittaker module is completely governed by the local function $\eta$.

Corollary 6.23**.**

Let $a,b\in\Bbbk$. Then the $\mathcal{O}$-module $\operatorname{Ind}^{\mathcal{O}}_{\mathcal{F}_{n}\mathcal{O}}\Bbbk_{\chi_{a,b}}$ is irreducible if and only if $\operatorname{Ind}^{\mathcal{O}}_{\mathcal{F}_{n}\mathcal{O}}\Bbbk_{\eta}$ is irreducible. Consequently, the universal Whittaker module $M_{\varphi_{n}}=\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathcal{O}}\Bbbk_{\chi}$ is irreducible if and only if $\operatorname{Ind}^{\mathcal{O}}_{\mathcal{F}_{n}\mathcal{O}}\Bbbk_{\chi_{1,1}}$ is irreducible. ∎

Remark 6.24**.**

Corollary 6.23 implies that reducibility of the universal Whittaker module of type $\varphi_{n}$ is equivalent to the reducibility of the universal Whittaker module of type $\varphi^{\prime}_{n}\coloneqq{\left.\kern-1.2pt\chi_{1,1}\vphantom{\big|}\right|_{\mathcal{F}_{n}\mathcal{O}}}$.

We are now ready to complete the proof of Theorem 6.6.

Proof of Theorem 6.6.

First, assume that $\lambda_{2}\neq 4\lambda_{0}$ (if $n=0$) or that $\lambda_{2n+1}\neq 4\lambda_{2n-1}$ or $\lambda_{2n+2}\neq 4\lambda_{2n}$ (if $n\geq 1$). By Corollary 6.23, $M_{\varphi_{n}}$ is irreducible if and only if $\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathcal{O}}\Bbbk_{\chi_{1,1}}$ is irreducible. Now, Proposition 6.16 implies the following:

• •

If $n=0$, then $\alpha_{1}\neq 0$.

• •

If $n\geq 1$, then $\alpha_{2n}$ and $\alpha_{2n+1}$ are not simultaneously 0.

By Corollary 6.20, we know that $\mathfrak{p}_{\chi_{1,1}}=\mathcal{F}_{n}\mathcal{O}$ is a polarisation of $\chi_{1,1}$. Furthermore, the order of each one-point local function which makes up $\chi_{1,1}$ is at least 1. Thus, Theorem 6.21 implies that $\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathbb{W}_{1}}\Bbbk_{\chi_{1,1}}$ is an irreducible $\mathbb{W}_{1}$-module. It is easy to see that

[TABLE]

so the irreducibility of $\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathbb{W}_{1}}\Bbbk_{\chi_{1,1}}$ as a $\mathbb{W}_{1}$-module implies the irreducibility of $\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathcal{O}}\Bbbk_{\chi_{1,1}}$ as an $\mathcal{O}$-module. We therefore conclude that that $M_{\varphi_{n}}$ is irreducible.

For the converse, the case $n=0$ follows from Lemma 6.7 and the case $n\geq 1$ follows from Lemma 6.9. ∎

6.3. Whittaker modules for the centreless BCCA

We move on to the study of Whittaker modules over the full centreless BCCA. Although these may seem more complicated, since the BCCA is a larger Lie algebra than $\mathcal{O}$, we will see that the conditions governing the reducibility of the universal Whittaker modules over $\mathfrak{b}$ are much simpler, and our proof is more elementary and direct. This is thanks to the abelian subalgebra $\mathcal{P}$ of $\mathfrak{b}$. As we did in Subsection 6.1, we start by defining Whittaker functions on $\mathfrak{b}$.

Definition 6.25**.**

A Whittaker function on $\mathfrak{b}$ is a Lie algebra homomorphism $\psi_{n}\colon\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$ for some $n\in\mathbb{N}$.

The following result is an immediate consequence of Lemma 5.5 and Proposition 5.7.

Lemma 6.26**.**

A Whittaker function $\psi_{n}\colon\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$ is determined by its values on $\mathcal{U}_{n},\dots,\mathcal{U}_{2n+2}$ and $\mathcal{V}_{n},\dots,\mathcal{V}_{2n}$. Once these values are determined, we have

[TABLE]

for all $m\geq 2n+1$. ∎

We now define the universal Whittaker modules for the full centreless BCCA, and then proceed to study their reducibility.

Definition 6.27**.**

Let $\psi_{n}\colon\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$ be a Whittaker function and $\Bbbk_{\psi_{n}}=\Bbbk 1_{\psi_{n}}$ be the one-dimensional $\mathcal{F}_{n}\mathfrak{b}$-module defined by $\psi_{n}$, in other words,

[TABLE]

for all $x\in\mathcal{F}_{n}\mathfrak{b}$. Define the universal Whittaker module for $\mathfrak{b}$ of type $\psi_{n}$ to be the induced module

[TABLE]

As in Remark 6.5, the $\mathfrak{b}$-modules $M_{\psi_{n}}$ for $n\geq 1$ are Whittaker modules, but $M_{\psi_{0}}$ are not Whittaker modules.

The following result is the main theorem of this subsection.

Theorem 6.28**.**

Let $n\in\mathbb{N}$, and let $\psi_{n}\colon\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$ be a Whittaker function. If $n=0$, then $M_{\psi_{0}}$ is irreducible if and only if $\psi_{0}(\mathcal{V}_{0})\neq 0$. If $n\geq 1$, then $M_{\psi_{n}}$ is irreducible if and only if $\psi_{n}(\mathcal{V}_{2n})\neq 0$ or $\psi_{n}(\mathcal{V}_{2n-1})\neq 0$.

We first prove the easier direction of Theorem 6.28: we construct a nonzero proper submodule of $M_{\psi_{n}}$ when the conditions of Theorem 6.28 are not satisfied. We first consider the case $n=0$.

Lemma 6.29**.**

Suppose $n=0$ and $\psi_{0}(\mathcal{V}_{0})=0$. Then $U(\mathfrak{b})\mathcal{V}_{-1}\cdot 1_{\psi_{0}}$ is a proper submodule of $M_{\psi_{0}}$.

Proof.

By Lemma 6.26, we have $\psi_{0}(\mathcal{V}_{m})=0$ for all $m\geq 0$. In other words, $\mathcal{V}_{m}\cdot 1_{\psi_{0}}=0$ for all $m\geq 0$. Let $w\coloneqq\mathcal{V}_{-1}\cdot 1_{\psi_{n}}$. It is clear that $\mathcal{V}_{m}\cdot w=0$ for all $m\geq 0$, since the $\mathcal{V}_{k}$ commute. Furthermore, for $m\geq 0$, we have

[TABLE]

since $m+1\geq 0$, where we used that $\mathcal{V}_{k}\cdot w=0$ for all $k\geq 0$. If $m\geq 1$, then the above implies that $\mathcal{U}_{m}\cdot w=\psi_{0}(\mathcal{U}_{m})w$. On the other hand, if $m=0$, then

[TABLE]

It follows that $\Bbbk w$ is a one-dimensional $\mathcal{F}_{0}\mathfrak{b}$-module, and thus $U(\mathfrak{b})\cdot w$ is a proper submodule of $M_{\psi_{0}}$. ∎

Remark 6.30**.**

In the notation from the proof of Lemma 6.29, it is easy to see that $U(\mathfrak{b})\cdot w$ is isomorphic to $M_{\psi_{0}^{\prime}}$, where $\psi_{0}^{\prime}\colon\mathcal{F}_{0}\mathfrak{b}\to\Bbbk$ is given by $\psi_{0}^{\prime}(\mathcal{U}_{0})=\psi_{0}(\mathcal{U}_{0})-4$, $\psi_{0}^{\prime}(\mathcal{U}_{m})=\psi_{0}(\mathcal{U}_{m})$ for $m\geq 1$, and $\psi_{0}^{\prime}(\mathcal{V}_{m})=0$ for $m\geq 0$. Therefore, Lemma 6.29 implies that there is chain of submodules

[TABLE]

Next, we construct an analogous submodule to the one in Lemma 6.29 in the case $n\geq 1$.

Lemma 6.31**.**

Let $n\geq 1$. If $\psi_{n}(\mathcal{V}_{2n})=\psi_{n}(\mathcal{V}_{2n-1})=0$, then $U(\mathfrak{b})\mathcal{V}_{n-1}\cdot 1_{\varphi_{n}}$ is a proper submodule of $M_{\psi_{n}}$ isomorphic to $M_{\psi_{n}}$ itself.

Proof.

Let $w\coloneqq\mathcal{V}_{n-1}\cdot 1_{\psi_{n}}$. For $m\geq n$, we have

[TABLE]

since $n+m-1\geq 2n-1$, where we used that $\mu_{k}=0$ for all $k\geq 2n-1$. Similarly,

[TABLE]

where we used that $\mathcal{V}_{m}$ and $\mathcal{V}_{n-1}$ commute.

The above shows that $\Bbbk w$ is a one-dimensional $\mathcal{F}_{n}\mathfrak{b}$-module isomorphic to $\Bbbk_{\psi_{n}}$. It is now easy to see that $U(\mathfrak{b})\cdot w$ is isomorphic to $M_{\psi_{n}}$. ∎

For the remainder of this section, we assume that at least one of $\psi_{n}(\mathcal{V}_{2n})$ and $\psi_{n}(\mathcal{V}_{2n-1})$ is nonzero (if $n\geq 1)$ or that $\psi_{0}(\mathcal{V}_{0})\neq 0$ (if $n=0$). As a result, we introduce the following notation.

Notation 6.32**.**

Fix $n\in\mathbb{N}$ and a Whittaker function $\psi_{n}\colon\mathcal{F}_{n}\mathfrak{b}\to\Bbbk$. For $m\geq n$, define

[TABLE]

If $n\geq 1$, let $\kappa\in\{2n-1,2n\}$ be maximal such that $\mu_{\kappa}\neq 0$. If $n=0$, let $\kappa=0$.

By Lemma 6.26, we have $\lambda_{m+2}=4\lambda_{m}$ and $\mu_{m}=0$ for all $m\geq 2n+1$.

Having introduced the necessary notation, we now proceed to prove the other direction of Theorem 6.28. To achieve this, we need to show that we can use actions of $\mathfrak{b}$ to get $1_{\varphi_{n}}$ from any nonzero element of $M_{\psi_{n}}$ when $\mu_{2n}\neq 0$ or $\mu_{2n-1}\neq 0$. We do this by successively removing $\mathcal{U}_{i}$ and $\mathcal{V}_{j}$ from elements of $U(\mathfrak{b})$. First, we prove that $\mathcal{V}_{m}$ annihilates elements in $M_{\psi_{n}}$ which do not contain $\mathcal{U}_{-1}$, provided $m>\kappa$. Recall that $\mathcal{P}=\operatorname{span}\{\mathcal{V}_{k}|k\geq-1\}$ is an abelian subalgebra of $\mathfrak{b}$.

Lemma 6.33**.**

Let $x=\mathcal{U}_{r_{1}}\dots\mathcal{U}_{r_{\ell}}v\in M_{\psi_{n}}$, where $v\in U(\mathcal{P})\cdot 1_{\psi_{n}}$ and $0\leq r_{1}\leq r_{2}\leq\dots\leq r_{\ell}\leq n-1$. Then $\mathcal{V}_{\kappa+m}\cdot x=0$ for all $m\geq 1$.

Proof.

The result is clearly true if $n=0$, so assume that $n\geq 1$. We proceed by induction on $\ell$, the base case $\ell=0$ being trivial. For the induction step, assume that the statement holds for some $\ell\in\mathbb{N}$. Let $x=\mathcal{U}_{r_{0}}\mathcal{U}_{r_{1}}\dots\mathcal{U}_{r_{\ell}}v$, where $v\in U(\mathcal{P})\cdot 1_{\psi_{n}}$ and $0\leq r_{0}\leq r_{1}\leq\dots\leq r_{\ell}\leq n-1$. For all $m\geq 1$, we have

[TABLE]

where we used the induction hypothesis in the last two equalities. ∎

Next, we extend Lemma 6.33 to the situation where the element of $M_{\psi_{n}}$ contains $\mathcal{U}_{-1}$.

Lemma 6.34**.**

Let $x\in U(\mathcal{F}_{0}\mathcal{O}\ltimes\mathcal{P})\cdot 1_{\psi_{n}}$. Then $\mathcal{V}_{\kappa+m}\cdot\mathcal{U}_{-1}^{N}x=0$ if $m>N$.

Proof.

Follows via an easy induction on $N$, with Lemma 6.33 as the base case. ∎

In the following result, we show that the action $\mathcal{V}_{\kappa+N}\cdot\mathcal{U}_{-1}^{N}x$ results in the complete removal of $\mathcal{U}_{-1}$, in other words, that $\mathcal{V}_{\kappa+N}\cdot\mathcal{U}_{-1}^{N}x$ is a nonzero element of $U(\mathcal{F}_{0}\mathcal{O}\ltimes\mathcal{P})\cdot 1_{\psi_{n}}$, where $x\in U(\mathcal{F}_{0}\mathcal{O}\ltimes\mathcal{P})\cdot 1_{\psi_{n}}$.

Proposition 6.35**.**

Let $x\in U(\mathcal{F}_{0}\mathcal{O}\ltimes\mathcal{P})\cdot 1_{\psi_{n}}\setminus\{0\}$. Then, for any $N\geq 1$, there exists $c\in\Bbbk\setminus\{0\}$ such that

[TABLE]

where lower degree terms means terms with lower degree than $x$ under the PBW filtration on $U(\mathfrak{b})$.

Proof.

We proceed by induction on $N$. First, consider the base case $N=0$:

[TABLE]

which is nonzero.

For the induction step, assume the statement is true for some $N\in\mathbb{N}$. Then

[TABLE]

where we used Lemma 6.34 in the last two equalities. By the induction hypothesis, it follows that $\mathcal{V}_{\kappa+N+1}\cdot\mathcal{U}_{-1}^{N+1}x\in U(\mathcal{F}_{0}\mathcal{O}\ltimes\mathcal{P})\setminus\{0\}$, which concludes the proof. ∎

Having shown how $\mathcal{U}_{-1}$ can be removed from any element of $U(\mathfrak{b})$, we now proceed to also remove $\mathcal{U}_{0}$.

Proposition 6.36**.**

Suppose $n\geq 1$. Then the following hold.

1. (1)

Let $x\in U(\mathcal{F}_{1}\mathcal{O}\ltimes\mathcal{P})\cdot 1_{\psi_{n}}\setminus\{0\}$. Then $\mathcal{V}_{\kappa}\cdot x=\mu_{\kappa}x$. 2. (2)

Let $x\in U(\mathcal{F}_{1}\mathcal{O}\ltimes\mathcal{P})\cdot 1_{\psi_{n}}\setminus\{0\}$. Then there exists $c\in\Bbbk\setminus\{0\}$ such that $(\mathcal{V}_{\kappa}-\mu_{\kappa})^{N}\cdot\mathcal{U}_{0}^{N}x=cx$.

Proof.

For (1), it suffices to consider $x=\mathcal{U}_{r_{1}}\dots\mathcal{U}_{r_{\ell}}y$, where $y\in U(\mathcal{P})\cdot 1_{\psi_{n}}$ and $1\leq r_{1}\leq r_{2}\leq\dots\leq r_{\ell}\leq n-1$. The result is clearly true for $n=1$, so assume that $n\geq 2$. We proceed by induction on $\ell$, the case $\ell=0$ being trivial. For the induction step, assume that the statement is true for some $\ell\in\mathbb{N}$. Then

[TABLE]

By the induction hypothesis, we have

[TABLE]

Therefore,

[TABLE]

By Lemma 6.33, it follows that $\Big((\kappa-r_{0}-1)\mathcal{V}_{\kappa+r_{0}+2}-4(\kappa-r_{0})\mathcal{V}_{\kappa+r_{0}}\Big)\mathcal{U}_{r_{1}}\dots\mathcal{U}_{r_{\ell}}y=0$, since $r_{0}\geq 1$. We conclude that $\mathcal{V}_{\kappa}\cdot\mathcal{U}_{r_{0}}\mathcal{U}_{r_{1}}\dots\mathcal{U}_{r_{\ell}}y=\mu_{\kappa}\mathcal{U}_{r_{0}}\mathcal{U}_{r_{1}}\dots\mathcal{U}_{r_{\ell}}y$, which concludes the proof of (1).

For (2), the proof is similar to that of Proposition 6.35, using (1). ∎

The next result shows how to remove all the other $\mathcal{U}_{i}$ from an element of $U(\mathfrak{b})$ by successively acting by $(\mathcal{V}_{m}-\mu_{m})$ for some appropriately chosen $m\in\mathbb{N}$. Of course, if $n=0$ or $n=1$, then we have already removed all $\mathcal{U}_{i}$ in the previous results, so we do not need to consider these cases.

Proposition 6.37**.**

Suppose $n\geq 2$, and let $x=\mathcal{U}_{q_{1}}^{p_{1}}\dots\mathcal{U}_{q_{k}}^{p_{k}}y$, where $y\in U(\mathcal{P})\cdot 1_{\psi_{n}}$ and $1\leq q_{1}<q_{2}<\dots<q_{k}\leq n-1$. Then

[TABLE]

Proof.

Rewrite $x$ as

[TABLE]

where $1\leq s_{1}\leq s_{2}\leq\dots\leq s_{\ell}\leq n-1$. Of course, we have $s_{1}=q_{1}$. For $m\geq n$, we have

[TABLE]

If $m+s_{j}>\kappa$, then the $j^{\text{th}}$ term of the above summation is zero, by Lemma 6.33. In particular, if $m>\kappa-s_{1}=\kappa-q_{1}$, then the entire summation vanishes, and thus $\mathcal{V}_{m}\cdot x=\mu_{m}x$, as required.

Now suppose $m=\kappa-s_{1}=\kappa-q_{1}$. By the above discussion, we can only get a contribution from the $j^{\text{th}}$ term of the above summation if $s_{j}=s_{1}$, since in this case we have $m+s_{j}=\kappa$. Therefore, switching back to the original notation for $x$, we get

[TABLE]

By Proposition 6.36(1), it follows that

[TABLE]

and thus

[TABLE]

which concludes the proof. ∎

The next result is a direct consequence of Propositions 6.35, 6.36, and 6.37.

Corollary 6.38**.**

Let $M$ be a nonzero submodule of $M_{\psi_{n}}$. Then $M$ contains a nonzero element in $U(\mathcal{P})\cdot 1_{\psi_{n}}$. ∎

Thanks to Corollary 6.38, we can simply focus on elements of $M_{\psi_{n}}$ not containing any $\mathcal{U}_{i}$. Thankfully, the removal of $\mathcal{V}_{j}$ is much easier, since the $\mathcal{V}_{j}$ commute with each other.

Lemma 6.39**.**

Let $y=\mathcal{V}_{q_{1}}^{p_{1}}\dots\mathcal{V}_{q_{k}}^{p_{k}}\cdot 1_{\psi_{n}}\in M_{\psi_{n}}$ with $-1\leq q_{1}<\dots<q_{k}\leq n-1$. Then

[TABLE]

Proof.

Rewrite $y$ as

[TABLE]

where $-1\leq s_{1}\leq s_{2}\leq\dots\leq s_{\ell}\leq n-1$ (and thus $s_{1}=q_{1}$). We have

[TABLE]

Suppose $m=\kappa-q_{1}=\kappa-s_{1}$. Then $m+s_{j}>\kappa$ if $s_{j}>s_{1}$. Therefore, we only get a contribution from the $j^{\text{th}}$ term in the above summation if $s_{j}=s_{1}$, since in this case we have $m+s_{j}=\kappa$. It follows that

[TABLE]

which concludes the proof. The case where $m>\kappa-q_{1}$ follows similarly. ∎

We are now ready to prove Theorem 6.28.

Proof of Theorem 6.28.

One direction of the statement is simply Lemmas 6.29 and 6.31. For the converse, assume that $\mu_{2n}\neq 0$ or $\mu_{2n-1}\neq 0$, and let $M$ be a nonzero submodule of $M_{\psi_{n}}$. Using Corollary 6.38, choose an element $w\in M\cap(U(\mathcal{P})\cdot 1_{\psi_{n}})\setminus\{0\}$. Now apply Lemma 6.39 to $w$ repeatedly to deduce that $M$ contains a nonzero scalar multiple of the generating vector $1_{\psi_{n}}$ of $M_{\psi_{n}}$. Therefore, $M=M_{\psi_{n}}$, which concludes the proof. ∎

Remark 6.40**.**

One may wonder about using the methods of [CGL*+*25] to prove Theorems 6.6 and 6.28. The main issue is that $\mathcal{F}_{n}\mathcal{O}$ and $\mathcal{F}_{n}\mathfrak{b}$ are not ideals of $\mathcal{O}$ and $\mathfrak{b}$ for $n\geq 0$, making it impossible to apply their results directly, since quasi-Whittaker functions must be defined on ideals of the Lie algebra (recall Definition 2.12). But $\mathcal{F}_{n}\mathcal{O}$ and $\mathcal{F}_{n}\mathfrak{b}$ are ideals of $\mathcal{F}_{0}\mathcal{O}$ and $\mathcal{F}_{0}\mathfrak{b}$, so one could get some irreducibility results by inducing from $\mathcal{F}_{n}\mathcal{O}$ or $\mathcal{F}_{n}\mathfrak{b}$ to $\mathcal{F}_{0}\mathcal{O}$ or $\mathcal{F}_{0}\mathfrak{b}$ and analysing the Whittaker annihilator, as is done in [CGL*+*25]. However, deducing the irreducibility of the universal Whittaker module $M_{\varphi_{n}}$ or $M_{\psi_{n}}$ from the irreducibility of $\operatorname{Ind}_{\mathcal{F}_{n}\mathcal{O}}^{\mathcal{F}_{0}\mathcal{O}}\Bbbk_{\varphi_{n}}$ or $\operatorname{Ind}_{\mathcal{F}_{n}\mathfrak{b}}^{\mathcal{F}_{0}\mathfrak{b}}\Bbbk_{\psi_{n}}$ is a non-trivial task which would require similar analysis to our proof of Theorem 6.28. For this reason, we opted for a more direct proof in the case of $\mathfrak{b}$, and a proof using the orbit method for $\mathcal{O}$.

7. Future work

In summary, this paper provides new insights into the algebraic structure of the BCCA and studies some of its representations, namely restrictions of representations of larger Lie algebras and Whittaker modules. We hope our work serves as a useful starting point for further exploration of the BCCA and the related mathematical and physical concepts that one would encounter. We conclude this paper by presenting some of these future research directions:

1. (1)

The BCCA can be obtained as the Lie algebra contraction of $\operatorname{Vir}$ (see [BCC*+*25b, Equation (8)]). Hence, one could study $\mathfrak{b}$-modules obtained as contractions of $\operatorname{Vir}$-modules. Contractions at the level of representations are essential in Carrollian physics [CGOR16], but there is still much that is unclear in this regard. For instance, the contractions of $\operatorname{Vir}\times\operatorname{Vir}$-modules built from “mode algebras” or free fields (i.e., $BC$-systems, Heisenberg algebra etc.) have been considered in the physics literature444However, the procedure by which these representations are contracted seems different to the one introduced in [CGOR16]. (see for example [HSXZ22, Sections 5.1.2 and 5.2], [BBC*+*20, Section 4] and [BBM*+*24, Section 3.2.1]). However, there are still technical subtleties (e.g., issues with “normal-ordering”) when it comes to contracting these infinite-dimensional representations that need to be addressed. Attempting to construct sensible representations of the BCCA as contractions of representations of the Virasoro algebra may help address some of these subtleties. They should also help guide us towards the quantisation of the tensionless open string since we expect such representations to describe the possible spectra of this string. It would be particularly interesting if representations constructed in this manner belong to one of the classes of $\widehat{\mathfrak{b}}$-modules discussed in this paper. 2. (2)

The above line of enquiry leads naturally to the consideration of restrictions of $\operatorname{Vir}\times\operatorname{Vir}$-modules that are vertex operator algebra (VOA) modules, such as the modules formed from the Heisenberg VOA and $BC$-systems (i.e., semi-infinite wedge representation). The restrictions will not be VOAs, but may still provide field-theoretic insights for the tensionless open string, possibly informing us about its spectrum, quantisation etc. Some of these $\operatorname{Vir}\times\operatorname{Vir}$-modules are well-known examples of highest weight modules, so one should expect that after carefully dealing with normal-ordering ambiguities, their restriction to $\widehat{\mathfrak{b}}$-modules does indeed give rise to the “almost free” modules discussed in Section 3. Overall, together with the work in Section 3, we expect these two strands of exploration to aid in understanding the commutative diagram given in Figure 7.1 at the level of Lie algebra representations while providing new field-theoretic realisations of the BCCA, which would be very informative in deducing the possible spectra of quantised tensionless open strings. 3. (3)

Proving Conjecture 3.18 could not only aid the search for field-theoretic realisations of the BCCA, but also result in the development of novel methods to check the (in)decomposability of Lie algebra modules and/or study representations of infinite-dimensional non $\mathbb{Z}$-graded Lie algebras. 4. (4)

Studying the restriction of the $\operatorname{Vir}\times\operatorname{Vir}$ VOA module realised in terms of two fermionic $BC$ systems, each of conformal weight $(2,-1)$, may help formulate generalised notions of semi-infinite cohomology that can be formulated for Lie algebras without integer-grading. This could be the first step in formulating a BRST quantisation procedure for the tensionless open string. 5. (5)

A similar programme to the one in this paper can be undertaken for higher spin BCCAs [BCC*+*25b, Equation (26)]. Following our notation in Section 2, the higher spin BCCAs form the one-parameter family of Lie algebras $\mathcal{O}\ltimes\mathcal{P}_{b}$, where $b$ is usually taken to be an integer in physics, though the programme undertaken in this paper could be done for arbitrary $b\in\mathbb{C}$. 6. (6)

One could compute the low-dimensional cohomology of the two-parameter family of higher spin BCCAs, analogous to what was done for the Lie algebras $\mathcal{W}(a,b)\coloneqq\mathcal{W}\ltimes I(a,b)$ (see Definition 2.2) in [GJP11]. 7. (7)

Finally, our work can be extended to the supersymmetric versions of the BCCA. Two such algebras, dubbed the homogenous and inhomogeneous boundary superconformal Carrollian algebras (BSCCAs), were obtained in [BCC*+*25a] via similar constructions to the ones in [BCC*+*25b]. The study of modules over the homogenous BSCCA should serve as an invaluable stepping stone towards understanding the spectrum of the tensionless open superstring.

Appendix A More subalgebras of the Witt algebra

Given the results of Subsection 4.2, it is natural to ask if we get similar subalgebras of the Witt algebra by choosing generators

[TABLE]

for some $\lambda,\mu\in\Bbbk^{*}$. As the following lemma shows, these elements generate the entirety of $\mathcal{W}$ unless $\mu=\lambda^{2}$.

Lemma A.1**.**

If $\mu\neq\lambda^{2}$, then $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$ generate $\mathcal{W}$.

Proof.

This proof is most easily done with the help of a computer. As such, we omit the tedious computations and instead focus on the main arguments of the proof.

Let $\mathfrak{g}$ be the subalgebra of $\mathcal{W}$ generated by $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$. For $n\geq 3$, inductively define $\mathcal{X}_{n}\coloneqq\frac{1}{n-2}[\mathcal{X}_{n-1},\mathcal{X}_{1}]\in\mathfrak{g}$. In this way, we have $\mathcal{X}_{n}=L_{n}+\text{lower degree terms}$ for all $n\geq 1$. There is another way to generate an element of degree 5: we have $\mathcal{Y}_{5}\coloneqq[\mathcal{X}_{3},\mathcal{X}_{2}]\in\mathfrak{g}$. It can be checked that

[TABLE]

so we define $\mathcal{Y}_{3}\coloneqq-\frac{3}{16\lambda}(\mathcal{X}_{5}-\mathcal{Y}_{5})\in\mathfrak{g}$. Similarly, we can check that

[TABLE]

Since we assumed that $\mu\neq\lambda^{2}$, we know that $\mathcal{X}_{3}-\mathcal{Y}_{3}\neq 0$ (in fact, $\mathcal{X}_{3}=\mathcal{Y}_{3}$ if and only if $\mu=\lambda^{2}$). Thus, define $\mathcal{Y}_{1}\coloneqq\frac{16\lambda}{45(\lambda^{2}-\mu)}(\mathcal{X}_{3}-\mathcal{Y}_{3})\in\mathfrak{g}$. Continuing this process, we have

[TABLE]

so we define $\mathcal{Y}_{-1}\coloneqq-\frac{15}{16\lambda}(\mathcal{X}_{1}-\mathcal{Y}_{1})\in\mathfrak{g}$. Specifically, we have

[TABLE]

We also have the following element of $\mathfrak{g}$:

[TABLE]

Notice that $\mathcal{W}_{\leq 1}\coloneqq\operatorname{span}\{L_{n}\mid n\leq 1\}$ is a subalgebra of $\mathcal{W}$ isomorphic to $\mathbb{W}_{1}$. By [Buz23, Lemma 4.7], it follows that $\mathcal{Y}_{-1}$ and $\mathcal{Z}$ generate a subalgebra of $\mathcal{W}_{\leq 1}$ finite codimension. Therefore, $\mathfrak{g}\cap\mathcal{W}_{\leq 1}$ has finite codimension in $\mathcal{W}_{\leq 1}$.

By a completely symmetric argument, we also get that $\mathfrak{g}\cap\mathbb{W}_{1}$ has finite codimension in $\mathbb{W}_{1}$. Therefore, $\mathfrak{g}$ has finite codimension in $\mathcal{W}$. But now Proposition 4.3 implies that

[TABLE]

for some $g\in\Bbbk[t,t^{-1}]$, and therefore $g$ must divide both $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$. Without loss of generality, we may assume that $g\in\Bbbk[t]$, $g$ is monic, and $g(0)\neq 0$. Note that $\mathcal{X}_{1}=-(t^{2}-\lambda)\partial$ and $\mathcal{X}_{2}=-t^{-1}(t^{4}-\mu)\partial$. But $\gcd(t^{2}-\lambda,t^{4}-\mu)=1$, since $\mu\neq\lambda^{2}$, and therefore $g=1$, since $g$ is a common divisor of $\mathcal{X}_{1}$ and $\mathcal{X}_{2}$. Now (A.1) gives that $\mathfrak{g}=\mathcal{W}$. ∎