Imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$-modules in category $\mathcal O^q_{\text{red,im}}$

TL;DR

This paper establishes the existence of imaginary crystal bases for all modules in a specific category of quantum affine algebra representations, extending previous results that were limited to certain modules.

Contribution

It generalizes the existence of imaginary crystal bases from reduced imaginary Verma modules to all modules in the category $ ext{O}^q_{ ext{red,im}}$ for $U_q( ext{sl}(2))$.

Findings

Existence of imaginary crystal bases for all modules in the category.

Extension of previous results from specific modules to the entire category.

Provides a foundation for further study of module structures in quantum affine algebras.

Abstract

Recently we defined imaginary crystal bases for $U_q(\widehat{\mathfrak{sl}(2)})$- modules in category $\mathcal O^q_{\text{red,im}}$ and showed the existence of such bases for reduced quantized imaginary Verma modules for $U_q(\widehat{\mathfrak{sl}(2)})$. In this paper we show the existence of imaginary crystal basis for any object in the category $\mathcal O^q_{\text{red,im}}$.

Equations

$$E_0,E_1,F_0,F_1,K_0^{\pm 1},K_1^{\pm 1},D^{\pm 1}$$

$$D{D}^{-1}=D^{-1}D=K_i{K}_i^{-1}=K_i^{-1}{K}_i=1,$$

$$E_i{F}_j-F_j{E}_i={\delta }_{ij}\frac{K_i-K_i^{-1}}{q-q^{-1}},$$

$$K_i{E}_i{K}_i^{-1}=q^2{E}_i,K_i{F}_i{K}_i^{-1}=q^{-2}{F}_i,$$

$$K_i{E}_j{K}_i^{-1}=q^{-2}{E}_j,K_i{F}_j{K}_i^{-1}=q^2{F}_j,i\ne j,$$

$$K_i{K}_j-K_j{K}_i=0,K_{i}D-D{K}_i=0,$$

$$D{E}_i{D}^{-1}=q^{{\delta }_{i,0}}{E}_i,D{F}_i{D}^{-1}=q^{-{\delta }_{i,0}}{F}_i,$$

$$E_i^3{E}_j-[3]E_i^2{E}_j{E}_i+[3]E_i{E}_j{E}_i^2-E_j{E}_i^3=0,i\ne j,$$

$$F_i^3{F}_j-[3]F_i^2{F}_j{F}_i+[3]F_i{F}_j{F}_i^2-F_j{F}_i^3=0,i\ne j,$$

$$D{D}^{-1}=D^{-1}D$$

$$[{\gamma }^{\pm \frac{1}{2}},u]$$

$$[h_k,h_l]$$

$$[h_k,K]$$

$$D{h}_k{D}^{-1}$$

$$D{x}_k^{\pm }{D}^{-1}$$

$$K{x}_k^{\pm }{K}^{-1}$$

$$[h_k,x_l^{\pm }]$$

$$x_{k+1}^{\pm }{x}_l^{\pm }$$

$$=q^{\pm 2}{x}_k^{\pm }{x}_{l+1}^{\pm }-x_{l+1}^{\pm }{x}_k^{\pm },$$

$$[x_k^{+},x_l^{-}]$$

$$\text{where }\sum _{k=0}^{\infty }\psi (k)z^{-k}$$

$$\sum _{k=0}^{\infty }\varphi (-k)z^k$$

$$E_0$$

$$E_1$$

$$K_0$$

$$\varphi (u)=\sum _{p\in \mathbb{Z}}\varphi (p)u^{-p},\psi (u)=\sum _{p\in \mathbb{Z}}\psi (p)u^{-p},x^{\pm }(u)=\sum _{p\in \mathbb{Z}}{x}_p^{\pm }{u}^{-p}$$

$$[\varphi (u),\varphi (v)]=0=[\psi (u),\psi (v)]$$

$$\varphi (u)x^{\pm }(v)\varphi (u{)}^{-1}=g(u{v}^{-1}{\gamma }^{\mp 1/2}{)}^{\pm 1}{x}^{\pm }(v)$$

$$\psi (u)x^{\pm }(v)\psi (u{)}^{-1}=g(v{u}^{-1}{\gamma }^{\mp 1/2}{)}^{\mp 1}{x}^{\pm }(v)$$

$$(u-q^{\pm 2}v)x^{\pm }(u)x^{\pm }(v)=(q^{\pm 2}u-v)x^{\pm }(v)x^{\pm }(u)$$

$$[x^{+}(u),x^{-}(v)]=(q-q^{-1}{)}^{-1}(\delta (u/v\gamma )\psi (v{\gamma }^{1/2})-\delta (u\gamma /v)\varphi (u{\gamma }^{1/2}))$$

$$g(r)=g_q(r)=g_{q^{-1}}(r)=\{\begin{array}{ll}{q}^2& \text{if}r=0\\ (1-q^{-4})q^{2(r+1)}=(q^4-1)q^{2(r-1)},& \text{if}r>0.\end{array}$$

$$x_k^{-}{x}_{k+1}^{-}=q^2{x}_{k+1}^{-}{x}_k^{-}$$

$$x_l^{-}{x}_{k+1}^{-}=q^2{x}_{k+1}^{-}{x}_l^{-}-x_k^{-}{x}_{l+1}^{-}+q^2{x}_{l+1}^{-}{x}_k^{-}$$

$$h^{+}=h_{+}^{(s_k)}:=h_{r_1}^{s_1}\cdots h_{r_k}^{s_l},h^{-}:=h_{-}^{(s_k)}=h_{-r_1}^{s_1}\cdots h_{-r_k}^{s_l}$$

$$\bar{P}$$

$${\bar{P}}_l$$

$$G_l=G_l^{1/q}:=\prod _{j=1}^{l-1}{g}_{q^{-1}}(v_j/v_l),G_l^q=\prod _{j=1}^{l-1}g(v_l/v_j)$$

$${\Omega }_{\psi }(u)=\sum _{l\in \mathbb{Z}}{\Omega }_{\psi }(l)u^{-l},{\Omega }_{\varphi }(u)=\sum _{l\in \mathbb{Z}}{\Omega }_{\varphi }(l)u^{-l}$$

$${\Omega }_{\psi }(u)(\bar{P}):$$

$${\Omega }_{\varphi }(u)(\bar{P}):$$

$$\bar{P}=x^{-}(v_1)\cdots x^{-}(v_k)=\sum _{n\in \mathbb{Z}}\sum _{\genfrac{}{}{0ex}{}{n_1,n_2,\dots ,n_k\in \mathbb{Z}}{n_1+\cdots +n_k=n}}{x}_{n_1}^{-}\cdots x_{n_k}^{-}{v}_1^{-n_1}\cdots v_k^{-n_k}$$

$$\psi$$

$$=\sum _{k\ge 0}\sum _{p\in \mathbb{Z}}\sum _{n_i\in \mathbb{Z}}{\gamma }^{k/2}\psi (k){\Omega }_{\psi }(p)(x_{n_1}^{-}\cdots x_{n_k}^{-})v_1^{-n_1}\cdots v_k^{-n_k}{u}^{-k-p}$$

Full text

1. Introduction

Consider the affine Lie algebra $\widehat{\mathfrak{g}}=\widehat{\mathfrak{sl}(2)}$ with Cartan subalgebra $\widehat{\mathfrak{h}}$. Let $\{\alpha_{0},\alpha_{1}\}$ be the simple roots, $\delta=\alpha_{0}+\alpha_{1}$ the null root and $\Delta$ the set of roots for $\widehat{\mathfrak{g}}$ with respect to $\widehat{\mathfrak{h}}$. Then we have a natural (standard) partition of $\Delta=\Delta_{+}\cup\Delta_{-}$ into set of positive and negative roots which is closed (i.e. $\alpha,\beta\in\Delta_{+}$ and $\alpha+\beta\in\Delta$ implies $\alpha+\beta\in\Delta_{+}$). Corresponding to this standard partition we have a standard Borel subalgebra from which we induce the standard Verma module. Let $S=\{\alpha_{1}+k\delta\ |\ k\in\mathbb{Z}\}\cup\{l\delta\ |\ l\in\mathbb{Z}_{>0}\}$. Then $\Delta=S\cup-S$ is another closed partition of the root system $\Delta$ which is not Weyl group conjugate to the standard partition. The classification of closed partitions of the root system for affine Lie algebras was obtained by Jakobsen and Kac [JK85, JK89], and independently by Futorny [Fut90, Fut92]. For the affine Lie algebra $\widehat{\mathfrak{g}}$ the partition $\Delta=S\cup-S$ is the only nonstandard closed partition which gives rise to a nonstandard Borel subalgebra. The Verma module $M(\lambda)$ with highest weight $\lambda$ induced by this nonstandard Borel subalgebra is called the imaginary Verma module for $\widehat{\mathfrak{sl}(2)}$. Unlike the standard Verma module, the imaginary Verma module $M(\lambda)$ contains both finite and infinite dimensional weight spaces.

For generic $q$, consider the associated quantum affine algebra $U_{q}(\widehat{\mathfrak{sl}(2)})$ ([Dri85], [Jim85]). Lusztig [Lus88] proved that the integrable highest weight modules of $\widehat{\mathfrak{sl}(2)}$ can be deformed to those over $U_{q}(\widehat{\mathfrak{sl}(2)})$ in such a way that the dimensions of the weight spaces are invariant under the deformation. Following the framework of [Lus88] and [Kan95], it was shown in ([CFKM97], [FGM98]) that the imaginary Verma modules $M(\lambda)$ can also be $q$-deformed to the quantum imaginary Verma modules $M_{q}(\lambda)$ in such a way that the weight multiplicities, both finite and infinite-dimensional, are preserved.

Lusztig [Lus90] from a geometric view point and Kashiwara [Kas91] from an algebraic view point introduced the notion of canonical bases (equivalently, global crystal bases) for standard Verma modules $V_{q}(\lambda)$ and integrable highest weight modules $L_{q}(\lambda)$. The crystal basis ([Kas90]) can be thought of as the $q=0$ limit of the canonical basis. An important ingredient in the construction of crystal basis by Kashiwara in [Kas91], is a subalgebra $\mathcal{B}_{q}$ of the quantum group which acts on the negative part of the quantum group by left multiplication. This subalgebra $\mathcal{B}_{q}$, which is called the Kashiwara algebra, played an important role in the definition of the Kashiwara operators which defines the crystal basis. The algebra $\mathcal{B}_{q}$ has been defined in greater generality for integrable cases in [Mas09]. In [CFM10] we constructed an analog of Kashiwara algebra, denoted by $\mathcal{K}_{q}$ for the imaginary Verma module $M_{q}(\lambda)$ for the quantum affine algebra $U_{q}(\widehat{\mathfrak{sl}(2)})$ by introducing certain Kashiwara-type operators. Then we proved that a certain quotient $\mathcal{N}_{q}^{-}$ of $U_{q}(\widehat{\mathfrak{g}})$ is a simple $\mathcal{K}_{q}$-module and gave a necessary and sufficient condition for a particular quotient $\bar{M}_{q}(\lambda)$ (called reduced imaginary Verma module) of $M_{q}(\lambda)$ to be simple. These results were generalized to all untwisted affine Lie algebras in [CFM14].

In [CFM17] we considered a category $\mathcal{O}^{q}_{\text{red,im}}$ of $U_{q}(\widehat{\mathfrak{sl}(2)})$-modules and defined a crystal-like basis which we called imaginary crystal basis for modules in this category. We showed that the reduced imaginary Verma modules $\bar{M}_{q}(\lambda)$ are simple objects in $\mathcal{O}^{q}_{\text{red,im}}$ and any module in $\mathcal{O}^{q}_{\text{red,im}}$ is a direct sum of reduced imaginary Verma modules for $U_{q}(\widehat{\mathfrak{sl}(2)})$. Then we proved the existence of imaginary crystal basis for a reduced imaginary Verma module $\bar{M}_{q}(\lambda)$. In this paper we prove the existence of imaginary crystal basis for any object in the category $\mathcal{O}^{q}_{\text{red,im}}$.

2. Quantum affine algebra $U_{q}(\widehat{\mathfrak{sl}(2)})$

Let $\mathbb{F}$ denote a field of characteristic zero and $q$ be an indeterminant (not a root of unity). The quantum affine algebra $U_{q}(\widehat{\mathfrak{sl}(2)})$ is the associative $\mathbb{F}(q^{1/2})$-algebra with 1 generated by

[TABLE]

with defining relations:

[TABLE]

where, $[n]=\frac{q^{n}-q^{-n}}{q-q^{-1}}$ and $i,j\in\{0,1\}$.

There is an alternative realization for $U_{q}(\widehat{\mathfrak{sl}(2)})$, due to Drinfeld [Dri85], which we need. Let $U_{q}$ be the associative algebra with $1$ over $\mathbb{F}(q^{1/2})$ generated by the elements $x^{\pm}_{k}$ ($k\in\mathbb{Z}$), $h_{l}$ ($l\in\mathbb{Z}\setminus\{0\}$), $K^{\pm 1}$, $D^{\pm 1}$, and $\gamma^{\pm\frac{1}{2}}$ with the following defining relations:

[TABLE]

The isomorphism between $U_{q}(\widehat{\mathfrak{sl}(2)})$ and $U_{q}$ is given by:

[TABLE]

If one uses the formal sums

[TABLE]

Drinfeld’s relations (2.3), (2.8)-(2.10) can be written as

[TABLE]

where $g(t)=g_{q}(t)=\sum_{k\geq 0}g(r)t^{k}$ is the Taylor series at $t=0$ of the function $(q^{2}t-1)/(t-q^{2})$ and $\delta(z)=\sum_{k\in\mathbb{Z}}z^{k}$ is the formal Dirac delta function.

Remark 2.1**.**

Writing $g(t)=g_{q}(t)=\sum_{r\geq 0}g(r)t^{r}$ we have

[TABLE]

Considering Serre’s relation with $k=l$, we get

[TABLE]

The product on the right side is in the correct order for a basis element. If $k+1>l$ and $k\neq l$ in  (2.9), then $k+1>l+1$ so that $k\geq l+1$, and thus we can write

[TABLE]

and then after repeating the above identity, we will eventually arrive at sums of terms that are in the correct order. This is the opposite ordering of monomials as we had previously.

3. $\Omega$-operators and the Kashiwara algebra $\mathcal{K}_{q}$

Let $\mathbb{N}^{\mathbb{N}^{*}}$ denote the set of all functions from $\{k\delta\,|\,k\in\mathbb{N}^{*}\}$ to $\mathbb{N}$ with finite support. Then we write

[TABLE]

for $f=(s_{m})\in\mathbb{N}^{\mathbb{N}^{*}}$ where $f(r_{m})=s_{m}$ and $f(t)=0$ for $t\neq r_{i},m\leq i\leq l$.

Let $\mathcal{N}_{q}^{-}$, be the subalgebra of $U_{q}(\widehat{\mathfrak{sl}(2)})$ generated by $\gamma^{\pm 1/2}$, and $x^{-}_{n}$, $n\in\mathbb{Z}$. Consider $x^{-}(v)=\sum_{n}x^{-}_{n}v^{-n}$ as a formal power series of left multiplication operators $x^{-}_{n}:\mathcal{N}_{q}^{-}\to\mathcal{N}_{q}^{-}$.

As in [CFM10], for fixed $k$ we set

[TABLE]

and

[TABLE]

where $G_{1}:=1$. Then we define a collection of operators $\Omega_{\psi}(k),\Omega_{\phi}(k):\mathcal{N}_{q}^{-}\to\mathcal{N}_{q}^{-}$, $k\in\mathbb{Z}$, in terms of the generating functions as follows.

[TABLE]

where

[TABLE]

Note that $\Omega_{\psi}(u)(1)=\Omega_{\phi}(u)(1)=0$. More generally, let us write

[TABLE]

Then

[TABLE]

while

[TABLE]

Thus for a fixed $m$ and $k$-tuple $(n_{1},\dots,n_{k})$ the sum

[TABLE]

must be finite. Hence

[TABLE]

for $k$ sufficiently large.

Proposition 3.1** ([CFM10], Prop. 4.0.3).**

We have

[TABLE]

The identities in Proposition 3.1 can be rewritten as

[TABLE]

which may be written out in terms of components as

[TABLE]

We also have by  (3.8)

[TABLE]

as operators on $\mathcal{N}_{q}^{-}$.

We can also write  (3.4) in terms of components and as operators on $\mathcal{N}_{q}^{-}$

[TABLE]

The sum on the right hand side turns into a finite sum when applied to an element in $\mathcal{N}_{q}^{-}$, due to  (3.3).

Proposition 3.2**.**

[CFM10]** There is a unique nondegenerate symmetric bilinear form $(\enspace,\enspace)$ defined on $\mathcal{N}^{-}_{q}$ satisfying

[TABLE]

For $\mathbf{m}=(m_{1},\dots,m_{n})$ set

[TABLE]

and define the length of such a Poincare-Birkhoff-Witt basis element to be $|\mathbf{m}|=n$.

Proposition 3.3**.**

[CFM17]** For $\mathbf{m}=(m_{1},\dots,m_{n})\in\mathbb{Z}^{n}$, and $\mathbf{k}=(k_{1},\dots,k_{l})\in\mathbb{Z}^{l}$, if $n>l$, then

[TABLE]

On the other hand if $n=l$ with

[TABLE]

we have

[TABLE]

The Kashiwara algebra $\mathcal{K}_{q}$ is defined to be the $\mathbb{F}(q^{1/2})$-subalgebra of $\text{End}\,(\mathcal{N}_{q}^{-})$ generated by $\Omega_{\psi}(m),x^{-}_{n},\gamma^{\pm 1/2}$, $m,n\in\mathbb{Z}$, $\gamma^{\pm 1/2}$. Then the $\gamma^{\pm 1/2}$ are central and the following relations (which are implied by  (3.14)) are satisfied

[TABLE]

[TABLE]

together with

[TABLE]

Then $\mathcal{N}_{q}^{-}$ is a simple $\mathcal{K}_{q}$-module ([CFM10], Theorem 7.0.14).

4. Quantized Imaginary Verma Modules and Category $\mathcal{O}^{q}_{\text{red,im}}$

We begin by recalling some basic facts for the affine Kac-Moody algebra $\widehat{\mathfrak{sl}(2)}$ over field $\mathbb{F}$ with generalized Cartan matrix $A=(a_{ij})_{0\leq i,j\leq 1}=\begin{pmatrix}2&-2\\ -2&\ 2\\ \end{pmatrix}$

and its imaginary Verma modules. We use notations in [Kac90].

[TABLE]

with Lie bracket relations

[TABLE]

for $x,y\in{\mathfrak{s}l}(2)$, $n,m\in\mathbb{Z}$, where $(\ ,\ )$ denotes the trace form on ${\mathfrak{s}l}(2)$. For $x\in{\mathfrak{s}l(2)}$ and $n\in\mathbb{Z}$, we write $x(n)$ for $x\otimes t^{n}$. We consider the usual basis for ${\mathfrak{s}l}(2)$:

[TABLE]

Then $\{e_{0}=f(1),e_{1}=e(0),f_{0}=e(-1),f_{1}=f(0),h_{0}=-h(0)+c,h_{1}=h(0),d\}$ generate $\mathfrak{g}$ and $\mathfrak{h}={\rm span}\{h_{0},h_{1},d\}$ is the Cartan subalgebra.

Let $\Delta$ denote the root system of $\widehat{\mathfrak{sl}(2)}$, $\{\alpha_{0},\alpha_{1}\}$ the simple roots, $\delta=\alpha_{0}+\alpha_{1}$ the minimal imaginary root , $\{\Lambda_{0},\Lambda_{1}\}$ fundamental weights and $P=\mathbb{Z}\Lambda_{0}\oplus\mathbb{Z}\Lambda_{1}\oplus\mathbb{Z}\delta$ the weight lattice. Then

[TABLE]

A subset $S$ of the root system $\Delta$ is called closed if $\alpha,\beta\in S$ and $\alpha+\beta\in\Delta$ implies $\alpha+\beta\in S$. The subset $S$ is called a *closed partition * of the roots if $S$ is closed, $S\cap(-S)=\emptyset$, and $S\cup-S=\Delta$ [JK85],[JK89],[Fut90],[Fut92]. The set

[TABLE]

is a closed partition of $\Delta$ and is $W\times\{\pm{1}\}$-inequivalent to the standard partition of the root system into positive and negative roots [Fut94].

Let ${\mathfrak{g}}_{\pm}^{(S)}=\sum_{\alpha\in S}\hat{\mathfrak{g}}_{\pm\alpha}$. We have that ${\mathfrak{g}}_{+}^{(S)}$ is the subalgebra generated by $e(k)$ $(k\in\mathbb{Z})$ and $h(l)$ $(l\in\mathbb{Z}_{>0})$ and ${\mathfrak{g}}_{-}^{(S)}$ is the subalgebra generated by $f(k)$ $(k\in\mathbb{Z})$ and $h(-l)$ $(l\in\mathbb{Z}_{>0})$. Since $S$ is a partition of the root system, the algebra has a direct sum decomposition:

[TABLE]

Let $U({\mathfrak{g}}_{\pm}^{(S)})$ be the universal enveloping algebra of ${\mathfrak{g}}_{\pm}^{(S)}$. Then, by the PBW theorem, we have

[TABLE]

where $U({\mathfrak{g}}_{+}^{(S)})$ is generated by $e(k)$ $(k\in\mathbb{Z})$, $h(l)$ $(l\in\mathbb{Z}_{>0})$, $U({\mathfrak{g}}_{-}^{(S)})$ is generated by $f(k)$ $(k\in\mathbb{Z})$, $h(-l)$ $(l\in\mathbb{Z}_{>0})$ and $U({\mathfrak{h}})$, the universal enveloping algebra of ${\mathfrak{h}}$. A $U({\mathfrak{g}})$-module $V$ is called a weight module if $V=\oplus_{\mu\in P}V_{\mu}$, where

[TABLE]

Any submodule of a weight module is a weight module. A $U({\mathfrak{g}})$-module $V$ is called an $S$-highest weight module with highest weight $\lambda\in P$ if there is a non-zero $v_{\lambda}\in V$ such that (i) $u^{+}\cdot v_{\lambda}=0$ for all $u^{+}\in U({\mathfrak{g}}_{+}^{(S)})\setminus\mathbb{F}^{*}$, (ii) $h\cdot v_{\lambda}=\lambda(h)v_{\lambda}$, $c\cdot v_{\lambda}=\lambda(c)v_{\lambda}$, $d\cdot v_{\lambda}=\lambda(d)v_{\lambda}$, (iii) $V=U(\hat{\mathfrak{g}})\cdot v_{\lambda}=U({\mathfrak{g}}_{-}^{(S)})\cdot v_{\lambda}$. An $S$-highest weight module is a weight module.

For $\lambda\in P$, let $I_{S}(\lambda)$ denote the ideal of $U(\mathfrak{g})$ generated by $e(k)$ $(k\in\mathbb{Z})$, $h(l)$ $(l>0)$, $h-\lambda(h)1$, $c-\lambda(c)1$, $d-\lambda(d)1$. Then $M(\lambda)=U(\mathfrak{g})/I_{S}(\lambda)$ is the imaginary Verma module of $\mathfrak{g}$ with highest weight $\lambda$. Imaginary Verma modules have many structural features similar to those of standard Verma modules, with the exception of the infinite-dimensional weight spaces [Fut94]. It was shown in [CFKM97] that the imaginary Verma modules $M(\lambda)$ for $\mathfrak{g}$ can be $q$-deformed to the quantum imaginary Verma module $M_{q}(\lambda)$ for $U_{q}(\mathfrak{g})$ in such a way that the weight multiplicities, both finite and infinite-dimensional, are preserved. Indeed for $\lambda\in P$ the quantum imaginary Verma module $M_{q}(\lambda)$ can be described as follows. Denote by $I^{q}(\lambda)$ the ideal of $U_{q}$ generated by $x^{+}(k)$, $k\in\mathbb{Z}$, $a(l),l>0$, $K^{\pm 1}-q^{\lambda(h)}1$, $\gamma^{\pm\frac{1}{2}}-q^{\pm\frac{1}{2}\lambda(c)}1$ and $D^{\pm 1}-q^{\pm\lambda(d)}1$. Then

[TABLE]

Theorem 4.1** ([CFKM97], Theorem 3.6).**

The imaginary Verma module $M_{q}(\lambda)$ is simple if and only if $\lambda(c)\neq 0$.

Suppose now that $\lambda(c)=0$. Then $\gamma^{\pm\frac{1}{2}}$ acts on $M_{q}(\lambda)$ by $1$. Consider the ideal $J^{q}(\lambda)$ of $U_{q}$ generated by $I^{q}(\lambda)$ and $a(l)$ for all $l$. Denote

[TABLE]

Then $\bar{M}_{q}(\lambda)$ is a homomorphic image of $M_{q}(\lambda)$ which is called the reduced quantized imaginary Verma module. The module $\bar{M}_{q}(\lambda)$ has a $P$-gradation:

[TABLE]

Theorem 4.2** ([CFM10]).**

Let $\lambda\in P$ be such that $\lambda(c)=0$. Then the module $\bar{M}_{q}(\lambda)$ is simple if and only if $\lambda(h)\neq 0$.

Consider the set $\mathfrak{h}^{*}_{red}:=\{\lambda\in\mathfrak{h}^{*}\,|\,\lambda(c)=0,\lambda(h)\neq 0\}$. Let $G_{q}$ be the quantized Heisenberg subalgebra generated by $h_{k},k\in\mathbb{Z}\setminus\{0\}$ and $\gamma$. We say that a nonzero $U_{q}({\mathfrak{g}})$-module $V$ is $G_{q}$-compatible if $V$ has a decomposition $V=TF(V)\oplus T(V)$ into a sum of nonzero $G_{q}$-submodules such that $G_{q}$ is bijective on $TF(V)$ (that any nonzero element $g\in G_{q}$ is a bijection on $TF(V)$) and $TF(V)$ has no nonzero $U_{q}(\mathfrak{g})$-submodule, and $G_{q}\cdot T(V)=0$.

The category $\mathcal{O}^{q}_{\text{red,im}}$ has as objects $U_{q}({\mathfrak{g}})$-modules $M$ such that

1. (1)

[TABLE] 2. (2)

$x^{+}_{n}$, $n\in\mathbb{Z}$ act locally nilpotently, 3. (3)

$M$ is $G_{q}$-compatible.

The morphisms in this category are the $U_{q}({\mathfrak{g}})$-module homomorphisms. Then for $M\in\mathcal{O}^{q}_{\text{red,im}}$ there exists $\lambda_{i}\in\mathfrak{h}^{*}_{red}$, $i\in I$, with $M\cong\bigoplus_{i\in I}\bar{M}_{q}(\lambda_{i})$ ([CFM17], Theorem 6.0.4).

For $M\in\mathcal{O}^{q}_{\text{red,im}}$, we can write $M=\oplus_{i}\bar{M}_{q}(\lambda_{i})$ with $\bar{M}_{q}(\lambda_{i})=\oplus\mathbb{F}(q^{1/2})x^{-}_{n_{1}}\cdots x^{-}_{n_{k}}v_{\lambda_{i}}$. We define $\tilde{\Omega}_{\psi}(m)$ and $\tilde{x}_{m}^{-}$ on each $\bar{M}_{q}(\lambda_{i})$ as in  (3.1):

[TABLE]

Hence the following result follows.

Theorem 4.3**.**

[CFM17]** The operators $\tilde{\Omega}_{\psi}(m)$ and $\tilde{x}_{m}^{-}$ are well defined on objects in the category $\mathcal{O}^{q}_{\text{red,im}}$. Moreover on each summand $\bar{M}_{q}(\lambda_{i})\cong\mathcal{N}_{q}^{-}$ they agree with the $\Omega_{\psi}(m)$ respectively left multiplication by $x_{m}^{-}$ defined as in  (3.1).

Theorem 4.4**.**

The operators $\tilde{\Omega}_{\psi}(m)$ and $\tilde{x}_{m}^{-}$ commute with $U_{q}(\mathfrak{g})$-module homomorphisms.

Proof.

It is enough to prove the statement for a $U_{q}(\mathfrak{g})$-module homomorphism $\nu:\bar{M}_{q}(\lambda)\to M$ from a highest weight module $\bar{M}_{q}(\lambda)$ to any module $M\in\mathcal{O}_{\text{ref},\text{im}}^{q}$ for some $\lambda\in\mathfrak{h}^{*}_{\text{red}}$. Now since $\bar{M}_{q}(\lambda)$ is simple, its image under $\nu$ is isomorphic to $\bar{M}_{q}(\lambda)$.

Consider now a basis element of $\bar{M}_{q}(\lambda)$ of the form $x^{-}_{n_{1}}\cdots x^{-}_{n_{k}}v_{\lambda}$. Then

[TABLE]

and

[TABLE]

∎

5. Imaginary crystal lattice and imaginary crystal basis

We first recall the definition of imaginary crystal lattice and imaginary crystal basis for modules in category $\mathcal{O}^{q}_{\text{red,im}}$. Let $\mathbb{A}_{0}$ to be the ring of rational functions in $q^{1/2}$ with coefficients in a field $\mathbb{F}$ of characteristic zero, regular at [math]. Let $\mathbb{A}=\mathbb{F}[q^{1/2},q^{-1/2},\frac{1}{[n]_{q}},n>1]$, and $\pi=\{-k\alpha_{1}+m\delta\,|\,k>0,m\in\mathbb{Z}\}\cup\{0\}$. Let $M\in\mathcal{O}^{q}_{\text{red,im}}$. We call a free $\mathbb{A}_{0}$-submodule $\mathcal{L}$ of $M$ an imaginary crystal $\mathbb{A}_{0}$-lattice of $M$ if the following hold

1. (i)

$\mathbb{F}(q^{1/2})\otimes_{\mathbb{A}_{0}}\mathcal{L}\cong M$, 2. (ii)

$\mathcal{L}=\oplus_{\lambda\in\pi}\mathcal{L}_{\lambda}$ and $\mathcal{L}_{\lambda}=\mathcal{L}\cap M_{\lambda}$, 3. (iii)

$\tilde{\Omega}_{\psi}(m)\mathcal{L}\subseteq\mathcal{L}$ and $\tilde{x}^{-}_{m}\mathcal{L}\subseteq\mathcal{L}$ for all $m\in\mathbb{Z}$.

An imaginary crystal basis of a $U_{q}({\mathfrak{g}})$-module $M\in\mathcal{O}^{q}_{\text{red,im}}$ is a pair $(\mathcal{L},\mathcal{B})$ satisfying

1. (1)

$\mathcal{L}$ is an imaginary crystal lattice of $M$, 2. (2)

$\mathcal{B}$ is an $\mathbb{F}$-basis of $\mathcal{L}/q\mathcal{L}\cong\mathbb{F}\otimes_{\mathcal{A}_{0}}\mathcal{L}$, 3. (3)

$\mathcal{B}=\sqcup_{\mu\in\pi}\mathcal{B}_{\mu}$, where $\mathcal{B}_{\mu}=\mathcal{B}\cap(\mathcal{L}_{\mu}/q\mathcal{L}_{\mu})$, 4. (4)

$\tilde{x}_{m}^{-}\mathcal{B}\subset\pm\mathcal{B}\cup\{0\}$ and $\tilde{\Omega}_{\psi}(m)\mathcal{B}\subset\pm\mathcal{B}\cup\{0\}$, 5. (5)

For $m\in\mathbb{Z}$, if $\tilde{\Omega}_{\psi}(-m)b\neq 0$ and $\tilde{x}_{m}^{-}b\neq 0$ for $b\in\mathcal{B}$, then $\tilde{x}_{m}^{-}\tilde{\Omega}_{\psi}(-m)b=\tilde{\Omega}_{\psi}(-m)\tilde{x}_{m}^{-}b$. .

For $\lambda\in\mathfrak{h}^{*}_{red}$, $\bar{M}_{q}(\lambda)\in\mathcal{O}^{q}_{\text{red,im}}$, define

[TABLE]

Then $\mathcal{L}(\lambda)$ is a imaginary crystal $\mathbb{A}_{0}$-lattice [CFM17] since

1. (i)

$\mathbb{F}(q^{1/2})\otimes_{\mathbb{A}_{0}}\mathcal{L}(\lambda)\cong\bar{M}_{q}(\lambda)$, 2. (ii)

$\mathcal{L}(\lambda)=\oplus_{\mu\in\pi}\mathcal{L}(\lambda)_{\mu}$ where $\mathcal{L}(\lambda)_{\mu}=\mathcal{L}(\lambda)\cap\bar{M}_{q}(\lambda)_{\mu}$, 3. (iii)

$\tilde{\Omega}_{\psi}(m)\mathcal{L}(\lambda)\subseteq\mathcal{L}(\lambda)$ and $\tilde{x}^{-}_{m}\mathcal{L}(\lambda)\subseteq\mathcal{L}(\lambda)$ for all $m\in\mathbb{Z}$.

Proposition 5.1**.**

[CFM17]** For $\lambda\in\mathfrak{h}^{*}_{red}$ we have

[TABLE]

For $\lambda\in\mathfrak{h}^{*}$ define

[TABLE]

Theorem 5.2**.**

[CFM17]** For $\lambda\in{\mathfrak{h}}^{*}_{red}$, the pair $(\mathcal{L}(\lambda),\mathcal{B}(\lambda))$ is an imaginary crystal basis of the reduced imaginary Verma module $\bar{M}_{q}(\lambda)$.

Now suppose $M\in\mathcal{O}^{q}_{\text{red,im}}$. Then there exists $\lambda_{i}\in\mathfrak{h}^{*}_{red}$, $i\in I$, with $M\cong\bigoplus_{i\in I}\bar{M}_{q}(\lambda_{i})$. Let $(\mathcal{L}(\lambda_{i}),\mathcal{B}(\lambda_{i}))$ be the imaginary crystal basis for $\bar{M}_{q}(\lambda_{i})$ for $i\in I$. Set $\mathcal{L}=\bigoplus_{i\in I}\mathcal{L}(\lambda_{i})$ and $\mathcal{B}=\bigsqcup_{i\in I}\mathcal{B}(\lambda_{i})$.

Theorem 5.3**.**

The pair $(\mathcal{L},\mathcal{B})$ is an imaginary crystal basis for $M\in\mathcal{O}^{q}_{\text{red,im}}$.

Proof.

First let us see that $\mathcal{L}$ is an imaginary crystal lattice:

1. (i)

$\mathbb{F}(q^{1/2})\otimes_{\mathbb{A}_{0}}\mathcal{L}\cong\oplus_{i\in I}\mathbb{F}(q^{1/2})\otimes_{\mathbb{A}_{0}}\mathcal{L}(\lambda_{i})\cong\oplus_{i\in I}\bar{M}_{q}(\lambda_{i})=M$, 2. (ii)

First we show that $\mathcal{L}_{\mu}=\left(\oplus_{i\in I}\mathcal{L}(\lambda_{i})\right)_{\mu}=\oplus_{i\in I}\mathcal{L}(\lambda_{i})_{\mu}$ where $\mathcal{L}(\lambda_{i})_{\mu}=\mathcal{L}(\lambda_{i})\cap M_{\mu}$. This follows because if $u\in\left(\oplus_{i\in I}\mathcal{L}(\lambda_{i})\right)_{\mu}$, then

[TABLE]

with $u_{i}\in\mathcal{L}(\lambda_{i})$ and $Ku=q^{\mu(h)}u$. Moreover since $\mathcal{L}(\lambda_{i})=\oplus_{\mu_{i}\in\pi}\mathcal{L}(\lambda_{i})_{\mu_{i}}$,

[TABLE]

with $u_{i,\mu_{i}}\in\mathcal{L}(\lambda_{i})$ and

[TABLE]

[TABLE]

Now since the sum $\oplus_{i\in I}\mathcal{L}(\lambda_{i})$ is direct, the above gives us

[TABLE]

for each $i$. Since $\mathcal{L}(\lambda_{i})=\oplus_{\mu_{i}\in\pi}\mathcal{L}(\lambda_{i})_{\mu_{i}}$ is a direct sum we have $\mu=\mu_{i}$ for all $i$. Thus $u_{i,\mu_{i}}\in\mathcal{L}(\lambda_{i})\cap M_{\mu}=\mathcal{L}(\lambda_{i})_{\mu}$. Finally we have $\mathcal{L}=\oplus_{i\in I}\mathcal{L}(\lambda_{i})=\oplus_{i\in I}\oplus_{\mu\in\pi}\mathcal{L}(\lambda_{i})_{\mu}=\oplus_{\mu\in\pi}\oplus_{i\in I}\mathcal{L}(\lambda_{i})_{\mu}=\oplus_{\mu\in\pi}\mathcal{L}_{\mu}$ and $\mathcal{L}_{\mu}=\mathcal{L}\cap M_{\mu}$. 3. (iii)

$\tilde{\Omega}_{\psi}(m)\mathcal{L}=\tilde{\Omega}_{\psi}(m)\left(\oplus_{i\in I}\mathcal{L}(\lambda_{i})\right)=\left(\oplus_{i\in I}\tilde{\Omega}_{\psi}(m)\mathcal{L}(\lambda_{i})\right)\subseteq\oplus_{i\in I}\mathcal{L}(\lambda_{i})=\mathcal{L}$ and $\tilde{x}^{-}_{m}\mathcal{L}=x^{-}_{m}\left(\oplus_{i\in I}\mathcal{L}(\lambda_{i})\right)=\left(\oplus_{i\in I}\tilde{x}^{-}_{m}\mathcal{L}(\lambda_{i})\right)\subseteq\oplus_{i\in I}\mathcal{L}(\lambda_{i})=\mathcal{L}$ for all $m\in\mathbb{Z}$.

This proves that $\mathcal{L}$ is an imaginary crystal lattice.

We know that $\mathcal{B}(\lambda_{i})$ is a $\mathbb{F}$-basis of $\mathcal{L}(\lambda_{i})/q\mathcal{L}(\lambda_{i})=\mathbb{F}\otimes_{\mathcal{A}_{0}}\mathcal{L}(\lambda_{i})$ for each $i\in I$. Now

[TABLE]

since $\mathcal{L}=\oplus_{i\in I}\mathcal{L}(\lambda_{i})$. Hence $\mathcal{L}/q\mathcal{L}$ has the $\mathbb{F}$-basis $\sqcup_{i\in I}\mathcal{B}(\lambda_{i})=\mathcal{B}$.

For (3) we have

[TABLE]

where $\mathcal{B}_{\mu}:=\sqcup_{i\in I}\mathcal{B}(\lambda_{i})_{\mu}$ and

[TABLE]

For (4) one notes that $\tilde{x}_{m}^{-}\mathcal{B}=\sqcup_{i\in I}\tilde{x}_{m}^{-}\mathcal{B}(\lambda_{i})\subseteq\pm\sqcup_{i\in I}\mathcal{B}(\lambda_{i})\cup\{0\}\subset\pm\mathcal{B}\cup\{0\}$ and similarly $\tilde{\Omega}_{\psi}(m)\mathcal{B}\subset\pm\mathcal{B}\cup\{0\}$,

For (5) we take $b\in\mathcal{B}$ and suppose $\tilde{\Omega}_{\psi}(-m)b\neq 0$ and $\tilde{x}_{m}^{-}b\neq 0$. Now $b\in\mathcal{B}(\lambda_{i})$ for some $i\in I$. Since $(\mathcal{L}(\lambda_{i}),\mathcal{B}(\lambda_{i}))$ is an imaginary crystal basis, we have $\tilde{x}_{m}^{-}\tilde{\Omega}_{\psi}(-m)b=\tilde{\Omega}_{\psi}(-m)\tilde{x}_{m}^{-}b$.

∎

We also have the following partial converse.

Theorem 5.4**.**

Let $M=M_{1}\oplus M_{2}$ where $M_{1}$ and $M_{2}$ are modules in the category $\mathcal{O}_{\text{red},\text{im}}^{q}$ and suppose $(\mathcal{L},\mathcal{B})$ is an imaginary crystal basis for $M$. Furthermore, suppose that there exists $\mathcal{A}_{0}$-submodules $\mathcal{L}_{j}\subset M_{j}$, and subsets $\mathcal{B}_{j}\subset\mathcal{L}_{j}/q\mathcal{L}_{j},\,j=1,2$ such that $\mathcal{L}=\mathcal{L}_{1}\oplus\mathcal{L}_{2}$ and $\mathcal{B}=\mathcal{B}_{1}\sqcup\mathcal{B}_{2}$. Then $(\mathcal{L}_{j},\mathcal{B}_{j})$ is an imaginary crystal basis of $M_{j},\,j=1,2$.

Proof.

It is straightforward to see that $\mathbb{F}(q^{1/2})\otimes_{\mathcal{A}_{0}}(\mathcal{L}_{j})_{\mu}\cong(M_{j})_{\mu}$ ($\mu\in\pi$), $\mathcal{L}_{j}=\mathcal{L}\cap M_{j}$ and $\mathcal{B}_{j}=\mathcal{B}\cap(\mathcal{L}_{j}/q\mathcal{L}_{j})$ for $j=1,2$ (see for instance [HK00, Theorem 4.2.10]).

Let $u\in\mathcal{L}_{\mu},$ for some $\mu\in\pi$. Then $u=u_{1}+u_{2}\in\mathcal{L}_{1}\oplus\mathcal{L}_{2}$ and

[TABLE]

Then $Ku_{1}-q^{\mu(h)}u_{1}=-Ku_{2}+q^{\mu(h)}u_{2}$ which must be zero since $\mathcal{L}\cap\mathcal{L}_{2}=\{0\}$. Thus $u_{j}\in\mathcal{L}_{j,\mu},\,j=1,2$. Hence $\mathcal{L}_{\mu}=\mathcal{L}_{1,\mu}\oplus\mathcal{L}_{2,\mu}$.

For $u\in\mathcal{L}_{j}\subset\mathcal{L}$, $u\in\oplus_{\mu\in\pi}\mathcal{L}_{\mu}$, so as a consequence $u=\sum_{\mu\in\pi}u_{\mu}$ with $u_{\mu}\in\mathcal{L}_{\mu}$ and we can write $u_{\mu}=u_{1,\mu}+u_{2,\mu}$ with $u_{j,\mu}\in(\mathcal{L}_{j})_{\mu}:=\mathcal{L}_{j}\cap M_{\mu}$. Consequently $u-\sum_{\mu\in\pi}u_{j,\mu}=\sum_{\lambda\in\pi}u_{k,\mu}\in\mathcal{L}_{j}\cap\mathcal{L}_{k}$ with $k\neq j$. Hence $u-\sum_{\mu\in\pi}u_{j,\mu}=0$ and we have $u\in\oplus_{\mu\in\pi}\mathcal{L}_{j,\mu}$.

Let $p_{j}:M\to M_{j}$ denote the natural projections which are $U_{q}(\mathfrak{g})$-module homomorphisms . Consider $u_{j}\in\mathcal{L}_{j}$. Since $\Omega_{\psi}(m)\mathcal{L}\subset\mathcal{L}$ and $\bar{x}_{m}^{-}\mathcal{L}\subset\mathcal{L}$ we write $\Omega_{\psi}(m)u_{1}=\bar{u}_{1}+\bar{u}_{2}$ and $\bar{x}_{m}^{-}u_{1}=\check{u}_{1}+\check{u}_{2}$ with $\check{u}_{j},\bar{u}_{j}\in\mathcal{L}_{j}$. By Theorem 4.4 we have

[TABLE]

and

[TABLE]

Hence $\Omega_{\psi}(m)(u_{1})\in\mathcal{L}_{1}$ and $\bar{x}_{m}^{-}(u_{1})\in\mathcal{L}_{1}$. Similarly $\Omega_{\psi}(m)(\mathcal{L}_{2})\subset\mathcal{L}_{2}$ and $\bar{x}_{m}^{-}(\mathcal{L}_{2})\subset\mathcal{L}_{2}$. This concludes the proof that $\mathcal{L}_{j}$ are imaginary crystal lattices.

We can write

[TABLE]

Using this isomorphism we have $\mathcal{B}_{j}=\mathcal{B}\cap(\mathcal{L}_{j}/q\mathcal{L}_{j})$ is an $\mathbb{F}$-basis of $\mathcal{L}_{j}/q\mathcal{L}_{j}\cong\mathbb{F}\otimes_{\mathcal{A}_{0}}\mathcal{L}_{j}$. Next we have

[TABLE]

and thus $\mathcal{B}_{j}=\sqcup_{\mu\in\pi}(\mathcal{B}_{j})_{\mu}$ where $(\mathcal{B}_{j})_{\mu}=\mathcal{B}_{j}\cap((\mathcal{L}_{j})_{\mu}/q(\mathcal{L}_{j})_{\mu})$.

Since the operators $\Omega_{\psi}(m)$ and $\bar{x}_{m}^{-}$ commute with $U_{q}(\mathfrak{g})$-module homomorphisms in the category $\mathcal{O}_{\text{red},\text{im}}^{q}$ (Theorem 4.4), we have that $\Omega_{\psi}(m)\mathcal{B}_{j}\subset\mathcal{B}_{j}\cup\{0\}$ and $\bar{x}_{m}^{-}\mathcal{B}_{j}\subset\mathcal{B}_{j}\cup\{0\}$ for all $m$ and $j=1,2$.

For $m\in\mathbb{Z}$, if $\tilde{\Omega}_{\psi}(-m)b\neq 0$ and $\tilde{x}_{m}^{-}b\neq 0$ for $b\in\mathcal{B}_{j}$ (and hence $b\in\mathcal{B}$), then $\tilde{x}_{m}^{-}\tilde{\Omega}_{\psi}(-m)b=\tilde{\Omega}_{\psi}(-m)\tilde{x}_{m}^{-}b$.

This completes the proof that $(\mathcal{L}_{j},\mathcal{B}_{j})$ is an imaginary crystal basis for $j=1,2$. ∎

Acknowledgement

The first author was partially support by a Simons Foundation Grant #319261. The second author was supported in part by the CNPq grant #301320/2013-6 and by the FAPESP grant #2014/09310-5. The third author was partially support by the Simons Foundation Grant #307555.