On Borel subalgebras of quantum groups

TL;DR

This paper classifies and analyzes Borel subalgebras of quantum groups, revealing new types and their module structures, with explicit classifications for low-rank cases and conjectures on their general shape.

Contribution

It introduces the concept of Borel subalgebras in quantum groups, identifies new examples, and studies their module theory and structural properties, including conjectures on their classification.

Findings

Identified all Borel subalgebras of U_q(sl_2) and U_q(sl_3).

Proved that induced modules from Borel subalgebras have all finite-dimensional irreducible modules as quotients.

Proposed and proved structural conjectures for the shape of triangular Borel subalgebras.

Abstract

For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example, ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We give two structural conjectures involving the associated graded right coideal subalgebra, which we prove in certain cases. In particular, they predict the shape of all triangular Borel subalgebras. As examples, we determine all Borel subalgebras of $U_q(\mathfrak{sl}_2)$ and $U_q(\mathfrak{sl}_3)$ and discuss the induced modules.

Equations

$$\bar{E}:=E{K}^{-1}+\lambda K^{-1},\bar{F}:=F+{\lambda }^{\prime }{K}^{-1},\text{for }\lambda {\lambda }^{\prime }=\frac{q^2}{(1-q^2)(q-q^{-1})}.$$

$$C=U_q^{-}[w_{-}{]}_{{\varphi }_{-}}\mathbb{k}[L]S(U_q^{+}[w]{)}_{{\varphi }_{+}}$$

$$\mathrm{gr}(B_{\lambda ,{\lambda }^{\prime }})=⟨K^{-1}⟩⟨S(E)⟩=U^{-}[1]\mathbb{C}[K^{-1}]S(U^{-}[s_1])$$

$${\Phi }^{+}(w)=\{\alpha \in {\Phi }^{+}\mid w^{-1}\alpha <0\}=\{{\beta }_i\mid i\in \{1,\dots ,\ell (w)\}\}$$

$$v<w\iff {\Phi }^{+}(v)\subset {\Phi }^{+}(w)$$

$$C=U^{+}[w]U^0{U}^{-}[v]$$

$$\mathrm{supp}(\varphi ):=\{\beta \in \Lambda \mid \exists x_{\beta }\in U_q^{-}[w]\text{ with }\varphi (x_{\beta })\ne 0\},$$

$$U^{-}[w{]}_{\varphi }:=\{\varphi (x^{(1)})x^{(2)}\mid \forall x\in U^{-}[w]\}$$

$$V(\chi )=U{\otimes }_{U^{+}{U}^0}{\mathbb{k}}_{\chi }\cong U^{-}{v}_{\lambda }$$

$$E_{\alpha }{v}_{\lambda }=0\text{ and }{K}_{\alpha }{v}_{\lambda }=\chi (K_{\alpha })v_{\lambda }\text{ for all }\alpha \in \Pi .$$

$$K{m}_i=\pm q^{n-2i}{m}_i$$

$$F{m}_i=\{\begin{array}{ll}{m}_{i+1},& \text{ for }i<n,\\ 0,& \text{ for }i=n\end{array}$$

$$E{m}_i=\{\begin{array}{ll}\pm [i{]}_q[n+1-i{]}_q{m}_{i-1},& \text{ for }i>0,\\ 0,& \text{ for }i=0\end{array}$$

$$C=C^{\le 0}{C}^{\ge 0}$$

$$\bar{E}:=E{K}^{-1}+\lambda K^{-1},\bar{F}:=F+{\lambda }^{\prime }{K}^{-1},\lambda {\lambda }^{\prime }=\frac{q^2}{(1-q^2)(q-q^{-1})}$$

$$[\bar{E},\bar{F}{]}_{q^2}$$

$$=\frac{K-K^{-1}}{q-q^{-1}}{q}^2{K}^{-1}+\lambda {\lambda }^{\prime }{K}^{-1}{K}^{-1}(1-q^2)$$

$$=\frac{q^2}{q-q^{-1}}\cdot 1$$

$$YX(Yv)=Y(qYX+1)v=(qt+1)Yv$$

$$YX(Xv)=\frac{1}{q}(XYX-X)v=\frac{1}{q}(t-1)Xv$$

$$XYv=(qYX+1)v=\left(\frac{1}{1-q}q+1\right)v=\frac{(q+1-q)}{1-q}v=tv$$

$$[E_{\alpha },F_{\alpha }{]}_1=\frac{K_{\alpha }-K_{\alpha }^{-1}}{q-q^{-1}}[E_{\alpha },K_{\alpha }{]}_{q^{-2}}=[F_{\alpha },K_{\alpha }{]}_{q^2}=0$$

$$B_{\lambda ,{\lambda }^{\prime }}:=⟨E{K}^{-1}+\lambda K^{-1},F+{\lambda }^{\prime }{K}^{-1}⟩$$

$$(E{K}^{-1}+{\lambda }_E{K}^{-1}).x_0$$

$$(F+{\lambda }^{\prime }{K}^{-1}).x_0$$

$$V_0=\{v\in V\mid h.v=ϵ(h)v\forall h\in T_{\alpha }(U^{-}[v])\}$$

$$X_{\mu }{F}_{\alpha }.v=q^{(\mu ,\alpha )}{F}_{\alpha }{X}_{\mu }.v+Y.v\text{ with some }Y\in T_{\alpha }(U^{-}[v])$$

$$V_0=\{v\in V\mid h.v=\chi (h)v\forall h\in T_{s_{\alpha }}(U^{-}[v])\}$$

$$B=U^{-}[v]U^{0}S(U^{+}[w])$$

$$V(B,\chi ):=U_q(\mathfrak{g}){\otimes }_B{\mathbb{k}}_{\chi }$$

$$(H{\otimes }_{B}M)\otimes V\cong H{\otimes }_B(M\otimes V)$$

$$h\otimes m\otimes v\mapsto h^{(1)}\otimes m\otimes S(h^{(2)}).v$$

$$V(B_{std},\chi ):=U_q(\mathfrak{g}){\otimes }_{B_0}{\mathbb{k}}_{\chi }\cong \mathbb{k}[E]1_{\chi }$$

Full text

Besides the positive part of the quantum group and its reflections, we find new unfamiliar Borel subalgebras, for example ones containing copies of the quantum Weyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modules and prove among others that they have all irreducible finite-dimensional modules as quotients. We then give structural results using the graded algebra, which in particular leads to a conjectural formula for all triangular Borel subalgebras, which we partly prove. As examples, we determine all Borel subalgebras of $U_{q}(\mathfrak{sl}_{2})$ and $U_{q}(\mathfrak{sl}_{3})$ and discuss the induced modules.

@setabstract

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Quantum groups

4. 2.2 Quantum group representation theory

5. 3 Borel subalgebras

6. 3.1 Main Definitions

7. 3.2 Example $A_{1}$

8. 3.3 Example: Homogeneous Borel subalgebras

9. 4 Induction of one-dimensional characters

10. 4.1 Definition and first properties

11. 4.2 Example $A_{1}$

12. 5 The structure of the graded algebra of a right coideal subalgebra

13. 5.1 Conjecture A

14. 5.2 Localization

15. 5.3 Induction step

16. 5.4 Three Criteria implying Conjecture A

17. 5.5 The proof of Conjecture A for type $A_{n},B_{n},C_{n},D_{n}$

18. 6 The graded algebra determining the representation theory

19. 6.1 Conjecture B

20. 6.2 Proof of Conjecture B in one direction

21. 7 Example $A_{2}$

22. 7.1 The Borel subalgebra $U^{-}[s_{1}]_{\phi_{-}}\Bbbk[(K_{1}K_{2}^{2})^{\pm 1}]S(U^{+}[s_{1}s_{2}s_{1}])_{\phi_{+}}$

23. 7.2 The Borel subalgebra $U^{-}[s_{1}s_{2}]_{\phi_{-}}\Bbbk[(K_{1}K_{2}^{2})^{\pm 1}]S(U^{+}[s_{1}s_{2}])_{\phi_{+}}$

24. 7.3 Classification Result

25. 7.4 Induction of one-dimensional characters

26. 8 Outlook

1. Introduction

Borel subalgebras are an essential element in the structure theory of a semisimple Lie algebra $\mathfrak{g}$, and the representations induced from a Borel subalgebra are an essential element in the representation theory of $\mathfrak{g}$. By Lie’s theorem, the Borel subalgebras in $\mathfrak{g}$ are precisely the maximally solvable subalgebras, and all Borel subalgebras are conjugate, in particular isomorphic.

I. Heckenberger has asked the analogous question for a quantum group $U_{q}(\mathfrak{g})$, namely to construct and classify those right coideal subalgebras $C$ of $U_{q}(\mathfrak{g})$ that are maximally solvable in the following sense: We call a right coideal subalgebra basic iff all its finite-dimensional irreducible representations are one-dimensional, and we call it Borel iff it is maximal with this property. It may be surprising that for quantum groups there are additional families of such Borel subalgebras, which are not present $\mathfrak{g}$. An example already for $\mathfrak{sl}_{2}$ and $q$ generic is given below.

The main motivation for studying Borel subalgebras of quantum groups is the more general goal to understand the set of all coideal subalgebras, which is considered a main problem in the area of quantum groups, with considerable progress made in [Let99, KS08, HS09, HK11a, HK11b]. Main examples are the quantum symmetric pairs, which are constructed in close analogy to the Lie algebra case, starting with [NS95, Let97]. For Lie algebras, Levi’s theorem states that any Lie subalgebra of $\mathfrak{g}$ decomposes into a solvable Lie algebra and a semisimple Lie algebra. While we have no such decomposition result for general coideal subalgebras, it seems promising to start with a classification of the two classes of coideal subalgebras corresponding to solvable and semisimple Lie subalgebras. In this article, we address the first problem, while [Beck16] considers the second.

Our second motivation is that the definition of a Borel subalgebra $C$ allows us to repeat the standard construction of Verma modules, namely inducing up a one-dimensional representation of $C$ to a representation of $U_{q}(\mathfrak{g})$. Our new Borel subalgebras lead to new families of quantum group representations, in which typically the Cartan part acts non-diagonalizably, but which otherwise share many features of usual Verma modules, for example having the finite-dimensional irreducible modules as quotients. For $\mathfrak{sl}_{2}$, these new modules have already appeared as operator-theoretic construction [Schm96] and in Liouville theory [Tesch01] Sec. 19. On the other hand, Futorny, Cox and collaborators, starting [Fut94, Cox94] have studied non-standard Borel subalgebras of affine Lie algebras; it is conceivable that our construction is related to this construction via Kazhdan-Lusztig correspondence.

The goal of this article is to initiate the study of these Borel subalgebras. For one, by constructing examples and give complete classifications for $\mathfrak{g}=\mathfrak{sl}_{2},\mathfrak{sl}_{3}$. Conceptually, we introduce structural tools and conjecture a general formula for Borel subalgebras, which may or may not be exhaustive. We can prove parts of this conjecture and check it in further examples. Moreover we prove general properties of the induced modules and construct then in example classes.

As a main structural tool for both questions, we propose to study the graded algebra $\mathrm{gr}(C)$ of a general coideal subalgebra $C$. For a coideal subalgebra $C$ in $U_{q}^{+}(\mathfrak{g})$, which where classified in [HK11b] as character-shifts, we give for $\mathrm{gr}(C)$ a conjectural formula (Conjecture A), which may be of independent interest and which also implies a formula for $\mathrm{gr}(C)$ for triangular $C$ in $U_{q}(\mathfrak{g})$. As a main result, we prove Conjecture A for all classical Lie algebras, i.e., for type $A_{n},B_{n},C_{n},D_{n}$.

We expect that this graded algebra essentially determines the representation theory of $C$. From these considerations we derive an explicit conjectural formula (Conjecture B) for all Borel subalgebras with a triangular decomposition. We are able to prove one direction of this assertion i.e. we obtain a necessary form for all basic triangular coideal subalgebras. For example, this shows that basic coideal subalgebras of $U_{q}(\mathfrak{sl}_{n+1})$ we constructed in [LV17] are maximal among all triangular basic coideal subalgebras, and that they have classificatory value.

We now discuss the content in more detail:

Our main object of study is the quantum group $U=U_{q}(\mathfrak{g})$, where $\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra and $q\in\Bbbk^{\times}$ is not a root of unity, and our base field $\Bbbk$ is an algebraically closed field of characteristic [math].

In Section 2, we collect preliminaries on quantum groups and their representation theory. We also briefly review the current state of the classification of right coideal subalgebras of quantum groups: In [HS09], the homogeneous graded right coideal subalgebras of $U_{q}^{\leq 0}(\mathfrak{g})$ are shown to be in bijection with Weyl group elements $w\in W$ via $U_{q}^{-}[w]U^{0}$. In [HK11a], the homogeneous graded right coideal subalgebras of $U_{q}(\mathfrak{g})$ are classified as suitable combinations $U_{q}^{-}[w_{-}]U^{0}S(U_{q}^{+}[w_{+}])$. In [HK11b], arbitrary right coideal subalgebras $C\subset U_{q}^{\leq 0}(\mathfrak{g})$, with $C^{0}$ a Hopf algebra, are classified to be so-called character-shifts $U_{q}^{-}[w]_{\phi}\Bbbk[L]$ with a suitable subalgebra of the Cartan part $\Bbbk[L]\subset U^{0}$. The classification of arbitrary right coideal subalgebras in $U_{q}(\mathfrak{g})$ is still a difficult open problem. We hope that by restricting our attention to basic right coideal subalgebras, we make a further step.

In Section 3, we give our definition of Borel subalgebras and discuss thoroughly two examples: First we classify all Borel subalgebras of $U_{q}(\mathfrak{sl}_{2})$. We find the two standard Borel subalgebras $U^{\geq 0}$ and $U^{\leq 0}$, which are already present in the Lie algebra $\mathfrak{sl}_{2}$, and we find a new family of non-homogeneous Borel subalgebras $B_{\lambda,\lambda^{\prime}}$:

Example** (3.6).**

In $U_{q}(\mathfrak{sl}_{2})$ there is a family of non-homogeneous Borel subalgebras $B_{\lambda,\lambda^{\prime}}$ with algebra generators

[TABLE]

Because of the denominator of $\lambda\lambda^{\prime}$, the subalgebras $B_{\lambda,\lambda^{\prime}}$ are not contained in the integral form of $U_{q}(\mathfrak{sl}_{2})$ and do not specialize to $q=1$. They interpolate between the two standard Borel subalgebras $U^{\geq 0}$ and $U^{\leq 0}$ for $\lambda\to 0$ and $\lambda^{\prime}\to 0$.

These Borel subalgebras are isomorphic to one another for different values of $\lambda$ and $\lambda^{\prime}$ by a Hopf automorphism of $U_{q}(\mathfrak{g})$ and they are isomorphic as an algebra to the quantum Weyl algebra $\langle x,y\rangle/(xy-q^{2}yx-1)$.

As a second initial class of examples, we show in Section 3.3 that, as expected, the Borel subalgebras of $U_{q}(\mathfrak{g})$ that are homogeneous with respect to the $\mathbb{N}^{n}$-grading are precisely the standard Borel subalgebras.

In Section 4, we turn our attention to representation theory: One important application of a Borel subalgebra $\mathfrak{b}$ of a Lie algebra $\mathfrak{g}$ is the construction of Verma modules $V(\lambda)$, which are induced modules $\mathfrak{g}\otimes_{\mathfrak{b}}\chi$ of a one-dimensional character on a Borel subalgebra $\chi:\mathfrak{b}\to\Bbbk$, which is given by a weight $\lambda\in\mathfrak{h}^{*}$.

Analogously, we construct an infinite-dimensional $U_{q}(\mathfrak{g})$-modules $V(B,\chi)$ for any Borel subalgebra $B$ in our sense and any one-dimensional character $\chi$. We then prove some general properties, in particular that for any given Borel subalgebra $B$ all finite-dimensional irreducible representations $L(\lambda)$ of $U_{q}(\mathfrak{g})$ appear as quotients of induced modules of one-dimensional character $\chi$ of $B$.

Then we study exemplary the induced modules for the unfamiliar Borel subalgebras $B_{\lambda,\lambda^{\prime}}$ in $U_{q}(\mathfrak{sl}_{2})$. As an algebra this is a quantum Weyl algebra, and one-dimensional representations are parametrized by complex numbers $e,f$ with $ef=\lambda\lambda^{\prime}$. The resulting infinite-dimensional $U_{q}(\mathfrak{sl}_{2})$-modules are isomorphic to the Cartan part $U^{0}=\Bbbk[K^{\pm 1}]$, and $U^{0}$ acts non-diagonalizably by left-multiplication. We determine all sub- and quotient modules. We find that for a generic character $\chi$ the induced module is irreducible, while for a discrete set of characters $\chi$ there is a unique non-trivial quotient, which is isomorphic to $L(\lambda)$.

In Section 5, we restrict our attention to triangular right coideal subalgebras $C=C^{\leq 0}C^{\geq 0}$. By the classification in [HK11b], both parts are character-shifts, so

[TABLE]

with unknown compatibility conditions for the data $(w_{\pm},\phi_{\pm},L)$. On the other hand $\mathrm{gr}(C)$ is (up to localization, see below) a coideal subalgebra classified in [HS09][HK11a] in terms of Weyl group elements $w_{\pm}^{\prime}$. The main question is to give a formula for $w_{\pm}^{\prime}$ depending on the data $(w_{\pm},\phi_{\pm},L)$. Motivated by the degree distribution and example calculations, we conjecture a general formula as Conjecture A.

The remainder of this section is devoted to derive different criteria that imply Conjecture A. By explicitly inspecting the root systems of type $A_{n},B_{n},C_{n},D_{n}$, where the root multiplicities are not too large, and the roots are given quite systematically, we find that our criteria imply Conjecture A for all coideal subalgebras in these cases, and quite trivially also for $G_{2}$. For $E_{6},E_{7},E_{8},F_{4}$ there are several exceptional cases, which are not covered by our criteria and which at our current state could only be settled by tedious computer calculations or (preferably) new structural ideas. We now sketch and discuss the structure of our proof:

• •

In Section 5.1, we formulate Conjecture A and list the consequences for arbitrary coideal subalgebras. We illustrate it by treating a first example in type $A_{2}$. A technical complication is that in general $\mathrm{gr}(C)^{0}$ is not a Hopf algebra (just a semigroup in the root lattice) so it need not be graded and [HK11a] does not apply. This is why we formulate our conjecture about the localization of $\mathrm{gr}(C)$ on $\mathrm{gr}(C)^{0}$, which captures the essential information.

• •

In Section 5.2, the localization is discussed. The main output of this consideration is Corollary 5.13, which compares the growth of the localizations and results in the formula $\ell(w)=\mathrm{rank}(\tilde{G})+\ell(w^{\prime})$.

• •

In Section 5.3, an inductive proof for Conjecture A is set up. For Weyl group elements $w=us_{\alpha_{i}}$ determining $C$, the Weyl group element determining the localization of $\mathrm{gr}(C)$ is prolonged to $w^{\prime}=u^{\prime}s_{\alpha_{j}}$ or $w^{\prime}=u^{\prime}$ depending on the case. It remains to show that $\alpha_{i}=\alpha_{j}$.

• •

In Section 5.4, we give three criteria that prove $\alpha_{i}=\alpha_{j}$ in the induction step. Criterion 1 uses the relation between $\alpha_{i}$ and $\alpha_{j}$ to determine all critical situations up to reflection. Criterion 2 proves the induction step provided $w$ does not have a unique ending. Criterion 3 proves the induction step by direct calculation of the character shift of a reflection $T_{s_{m}}$, provided $\alpha_{m}$ appears in the root with multiplicity zero. This calculation uses a slight generalization of [Jan96] 8.14 (6) from the case of a simple root to a root with multiplicity zero.

• •

In Section 5.5 we combine these criteria to cover all cases in $A_{n},B_{n},C_{n},D_{n}$.

For example for $B_{\lambda,\lambda^{\prime}}$ above, the graded algebra is as predicted by Conjecture A with $w^{\prime}=1$ for $w=s_{1}$:

[TABLE]

In Section 6 we start to apply Conjecture A to control representation-theoretic properties of a triangular right coideal subalgebra $C$. We formulate as Conjecture B a complete description of the data $(w_{\pm},\phi_{\pm},L)$ that lead to triangular Borel subalgebras of $U_{q}(\mathfrak{g})$, namely precisely those for which $\mathrm{gr}(C)$ localizes to a Borel subalgebra. We are only able to prove one direction of our conjecture by using Conjecture A (where proven). Our method to prove a coideal subalgebra is not basic is to find higher-dimensional $C$-representations as composition factors in the restriction of a suitable $U_{q}(\mathfrak{g})$-representation $L(\lambda)$.

In Section 7 we thoroughly treat the case $U_{q}(\mathfrak{sl}_{3})$, and we give a complete classification of all Borel subalgebras. We find three types of Borel subalgebras, which can all be given in terms of generators and relations:

• •

The standard Borel subalgebras.

• •

Borel subalgebras that come from the Borel subalgebras $B_{\lambda,\lambda^{\prime}}$ in $U_{q}(\mathfrak{sl}_{2})$, together with a remaining standard Borel subalgebra. This is the smallest example of a family we constructed [LV17].

• •

Another family of Borel subalgebras that consist of an extension of a Weyl algebra by another Weyl algebra.

In all these cases we also determine the induced representations.

Acknowledgement**.**

We are very thankful to Istvan Heckenberger for giving very valuable impulses at several stages of this paper. We are also thankful for his hospitality in Marburg, which was made possible by the Humboldt Institut Partnerschaft. We thank Yorck Sommerhäuser for comments and literature hints in the final stage of this article. The first author also receives partial support from the RTG 1670.

2. Preliminaries

2.1. Quantum groups

Throughout this article $\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra of rank $n$. Associated to this datum is a root system.

We denote a fixed set of positive simple roots by $\Pi=\{\alpha_{1},\ldots\alpha_{n}\}$ and the corresponding set of all positive roots by $\Phi^{+}$. The simple roots are a basis of the root lattice $\Lambda$ with bilinear form $(,)$ and Cartan matrix $c_{ij}=2\frac{(\alpha_{i},\alpha_{j})}{(\alpha_{i},\alpha_{i})}$. Let $W$ be the finite Weyl group acting on the lattice $\Lambda$ by reflections, generated by the simple reflections $s_{i}$ on the hyperplanes $\alpha_{i}^{\perp}$.

Let $\Bbbk$ be an algebraically closed field of characteristic [math], and let $q\in\Bbbk^{\times}$ be not a root of unity. We consider the quantum group $U=U_{q}(\mathfrak{g})$, and for the following standard facts we refer the reader e.g. to [Jan96]. The algebra $U$ is a deformation of the universal enveloping algebra of the Lie algebra $U(\mathfrak{g})$. It is generated by elements $E_{\alpha},F_{\alpha}$ and $K_{\alpha}^{\pm 1}$ with $\alpha\in\Pi$ and graded by $\Lambda$.

There is a triangular decomposition $U=U^{+}U^{0}U^{-}$ into subalgebras $U^{+},U^{0},U^{-}$ generated by the $E,K,F$, respectively, and we have Hopf subalgebras $U^{\geq 0}=U^{0}U^{+}$ and $U^{\leq 0}=U^{0}U^{-}$ corresponding to Borel subalgebras of $\mathfrak{g}$. The Cartan part ${U^{0}=\mathbb{C}[K_{\alpha_{1}}^{\pm 1},\ldots,K_{\alpha_{n}}^{\pm 1}]}$ is the group ring of the root lattice $\Lambda\cong\mathbb{Z}^{n}$.

The algebra $U$ can be endowed with the structure of a Hopf algebra, and this endows the category of representations $\mathrm{Rep}(U)$ of the quantum group with a tensor product $\otimes_{\Bbbk}$. We write $[X,Y]_{\lambda}=XY-\lambda YX$ for any $\lambda\in\Bbbk$ and $X,Y\in U_{q}(\mathfrak{g})$.

For every Weyl group element $w\in W$ there is an algebra automorphisms $T_{w}$ on $U_{q}(\mathfrak{g})$ due to Lusztig. For any fixed choice of a reduced expression of the longest element of the Weyl group $w_{0}=s_{\alpha_{i_{1}}}\cdots s_{\alpha_{i_{\ell(w_{0})}}}$ with $\ell(w_{0})=|\Phi^{+}|$ we get an enumeration of the set of positive roots $\beta_{k}=s_{\alpha_{i_{1}}}\cdots s_{\alpha_{i_{k-1}}}\alpha_{i_{k}}$. Using Lusztig’s automorphisms this defines root vectors $E_{\beta_{k}}=T_{s_{\alpha_{i_{1}}}\cdots s_{\alpha_{i_{k-1}}}}(E_{\alpha_{i_{k}}})$ and a PBW basis for $U^{+}$ consisting of sorted monomials in the root vectors, see [Jan96] Chapter 8.

Definition 2.1**.**

A subalgebra $C$ of a Hopf algebra $H$ is called a right coideal subalgebra if $\Delta(C)\subset C\otimes H$.

We call a right coideal subalgebra $C\subset U_{q}(\mathfrak{g})$ homogeneous if $U^{0}\subset C$. In particular $C$ is then homogeneous with respect to the $\Lambda$-grading.

Example 2.2**.**

Any Hopf subalgebra is in particular a right coideal subalgebra. For example $U^{+}U^{0}$ resp. $U^{-}U^{0}$ generated by the $E_{\alpha}$ resp. $F_{\alpha}$ and $K_{\alpha},K_{\alpha}^{-1}$ are homogeneous right coideal subalgebras. The subalgebras $U^{-}$ and $S(U^{+})$ are right coideal subalgebras, but $U^{+}$ is not.

Essential results in the theory of coideal subalgebras of quantum groups are:

Theorem 2.3** ([HS09] Theorem 7.3).**

For every $w\in W$ there is a right coideal subalgebra $U^{+}[w]U^{0}$, where $U^{+}[w]$ is generated by the root vectors $E_{\beta_{i}}$ for all $\beta_{i}$ in the following subset of roots

[TABLE]

In particular $|\Phi^{+}(w)|=\ell(w)$, and

[TABLE]

The space $U^{+}[w]$ does not depend on the choice of a reduced expression of $w$.

Conversely, every homogeneous right coideal subalgebra $C\subset U_{q}^{+}(\mathfrak{g})U^{0}$ is of this form for some $w$.

Theorem 2.4** ([HK11a] Theorem 3.8).**

Every homogeneous right coideal subalgebra $C\subset U_{q}(\mathfrak{g})$ is of the form

[TABLE]

for a certain subset of pairs $v,w\in W$.

Non-homogeneous right coideal subalgebras are only classified in $U^{\pm}U^{0}$:

Theorem 2.5** ([HK11b] Theorem 2.15).**

For $w\in W$, let $\phi:U^{-}[w]\to\Bbbk$ be a one-dimensional character and define

[TABLE]

which consists of mutually orthogonal simple roots. Take any subgroup $L\subset\mathrm{supp}(\phi)^{\perp}$, then there exists a character-shifted right coideal subalgebra

[TABLE]

and a right coideal subalgebra $U^{-}[w]_{\phi}\Bbbk[L]$ with group ring $\Bbbk[L]=\Bbbk[L]\subset U^{0}$.

Conversely, every right coideal subalgebra $C\subset U^{-}U^{0}$ with $C^{0}$ a Hopf algebras is of this form for some datum $(w,\phi,L)$.

2.2. Quantum group representation theory

A main idea for Lie algebra and quantum group representations is to take a large commutative subalgebra $\mathfrak{h}\subset\mathfrak{g}$ (Cartan part) and simultaneously diagonalize the action. The possible eigenvalues $\mathfrak{h}\to\Bbbk$ are called weights.

Definition 2.6** (Verma module).**

Every one-dimensional character $\chi$ of $U^{0}=\Bbbk[\Lambda]$ can be extended trivially to $U^{+}U^{0}$. Consider the induced $U$-representation

[TABLE]

generated by a highest weight vector $v_{\lambda}:=1\otimes 1_{\chi}$ with

[TABLE]

*The module $V(\chi)$ has a unique irreducible quotient module $L(\chi)$.

In particular for a weight $\lambda$ of $\mathfrak{g}$, the Verma module $V(\lambda)$ of type $+1$ is the induced representation associated to the character $\chi(K_{\alpha})=q^{(\lambda,\alpha)}$. If $\lambda$ is an integral dominant weight then $L(\lambda)$ is finite-dimensional. Every finite-dimensional irreducible module is the quotient of the induced module for a unique character $\chi$.

Recall that we always assume $q$ is not a root of unity. In this case, the category of finite-dimensional $U_{q}(\mathfrak{g})$-modules is semisimple, and resembles the case $U(\mathfrak{g})$.

Example 2.7**.**

For every $n\geq 0$ there are two irreducible $U_{q}(\mathfrak{sl}_{2})$-modules $L(n,\pm)$ of dimension $n+1$ with basis $m_{0},m_{1},\ldots,m_{n}$ such that

[TABLE]

Every finite-dimensional irreducible $U_{q}(\mathfrak{sl}_{2})$-module is of this form.

3. Borel subalgebras

3.1. Main Definitions

The main objects we wish to study in this article are:

Definition 3.1**.**

We call an algebra basic if all its finite-dimensional irreducible representations are one-dimensional.

Example 3.2**.**

Finite dimensional algebras are basic if they are basic in the usual sense i.e. $A/\mathrm{Rad}(A)\cong\Bbbk^{n}$. Commutative algebras are examples of basic algebras. Universal enveloping algebras of solvable Lie algebras are basic.

Definition 3.3**.**

We call a right coideal subalgebra of the Hopf algebra $U_{q}(\mathfrak{g})$ a Borel subalgebra if it is basic and it is maximal (with respect to inclusion) among all basic right coideal subalgebras.

The idea of this notion is to generalizes the characterization of a Borel subalgebra as the maximal solvable Lie subalgebra. A big simplification is the following additional technical assumption:

Definition 3.4**.**

We call a right coideal subalgebra $C$ of $U_{q}(\mathfrak{g})$ triangular, if

[TABLE]

with $C^{\leq 0}=C\cap U^{\leq 0}$ and $C^{\geq 0}=C\cap U^{\geq 0}$.

There are many examples of non-triangular right coideal subalgebras, but we have no example of a non-triangular Borel subalgebra.

Example 3.5**.**

All homogeneous Borel subalgebras are isomorphic to the standard Borel subalgebra $U^{+}U^{0}$. We will prove this in Section 3.3.

Example 3.6**.**

In $U_{q}(\mathfrak{sl}_{2})$ there is a family of non-homogeneous Borel subalgebras $B_{\lambda,\lambda^{\prime}}$ with algebra generators

[TABLE]

We will see below that $B_{\lambda,\lambda}$ is basic, because it is isomorphic to the quantum Weyl algebra. The maximality will be proven in the next section.

The condition on $\lambda,\lambda^{\prime}$ becomes clear if we calculate

[TABLE]

while for other choices $\lambda,\lambda^{\prime}$ we have $K^{-2}$ in the subalgebra $B_{\lambda,\lambda^{\prime}}$, and hence essentially all of $U_{q}(\mathfrak{g})$, which is surely not basic. On the other hand it is known:

Lemma 3.7**.**

The quantum Weyl algebra $\langle X,Y\rangle/(XY-q^{2}YX-c1)$ is basic.

Proof.

Let $V$ be a finite dimensional irreducible representation. Consider the eigenvector $v$ of the element $T:=YX$ with eigenvalue $t$ (which is related to the quantum Casimir). One can easily see, that $Yv$ is an eigenvector with the eigenvalue $qt+1$:

[TABLE]

Similarly, one can show that $Xv$ is an eigenvector of $T$ with eigenvalue $\frac{1}{q}(t-1)$:

[TABLE]

Thus the eigenvectors of $T$ are a basis of $V$, as $V$ is irreducible. On the other hand for each $i$ there are eigenvectors $Y^{i}v$ of $T$. As $V$ is finite dimensional, they cannot have pairwise distinct eigenvalues. Besides the possibility of $X,Y$ acting by zero, which is absurd, the only appearing eigenvalue has to be the fixed-point $t=\frac{1}{1-q}$. With this $t$ we get:

[TABLE]

As $T$ has only the eigenvalue $t$, it acts as a scalar on $V$, the same is true for $XY$. Thus $X$ and $Y$ commute on all of $V$, if it is irreducible, and thus each finite dimensional irreducible representation is one-dimensional, as asserted. ∎

3.2. Example $A_{1}$

The root system $A_{1}$ of rank $1$ has $\Phi=\{\alpha,-\alpha\}$ with ${(\alpha,\alpha)=2}$. The corresponding quantum group $U_{q}(\mathfrak{sl}_{2})$ is generated by $E,F,K,K^{-1}$ such that

[TABLE]

Theorem 3.8**.**

The Borel subalgebras of $U_{q}(\mathfrak{sl}_{2})$ are

• •

The standard Borel algebras $U^{\geq 0}$ and $U^{\leq 0}$.

• •

For any pair of scalars with $\lambda\lambda^{\prime}=\frac{q^{2}}{(1-q^{2})(q-q^{-1})}$ the algebra in Example 3.6

[TABLE]

Proof.

We know from the previous lemma that all algebras in the assertion are basic and we know (or check immediately) that they are right coideal subalgebras. The classification and the maximality are more difficult issues:

We know from [Vocke16] Theorem 4.11 and Lemma 3.5 that any right coideal subalgebra in $U_{q}(\mathfrak{sl}_{2})$ has a set of generators of the form $K^{i},\;EK^{-1}+\lambda_{E}K^{-1},\;F+\lambda_{F}K^{-1},\;EK^{-1}+c_{F}F+c_{K}K^{-1}$ for some constants $c_{F},c_{K},\lambda_{F},\lambda_{E}\neq 0$. We now check for all combinations of two such elements which algebra they roughly generate:

• [TABLE]

Most entries in the table follow simply by subtracting suitable multiples of one another (and ignoring possible $K$-powers). The entry involving $EK^{-1}+c_{F}F+c_{K}K^{-1}$ and $K^{i}$ comes from the fact that $K$ commutes differently with $E,F$, so suitable commutators return the individual summands, again up to $K$-powers.

Clearly $U$ is not basic. When the elements generate some $B_{\lambda,\lambda^{\prime}}$ but fail the condition on $\lambda\lambda^{\prime}$, then as we saw their commutator $[EK^{-1}+\lambda K^{-1},F+\lambda^{\prime}K^{-1}]_{q^{2}}$ is a linear combination of $1$ and $K^{-2}$ with non-zero coefficients, so the generated algebra contains $K^{-2}$. This situation is very similar to $U$, and indeed since every irreducible $U$-module $L(k,\pm)$ restricts to an irreducible over this algebra, thus we obtain irreducible higher-dimensional representations.

So the only remaining cases are when these two generators are equal, or when they generate $U^{\geq 0},U^{\leq 0}$ or when they generate $B_{\lambda,\lambda^{\prime}}$. ∎

Example 3.9**.**

*In upcoming proofs we will frequently use restrictions of suitable $U_{q}(\mathfrak{g})$-modules to construct higher-dimensional representations of some $B$ in question. The restrictions are in general neither irreducible nor semisimple. For a Borel subalgebra all composition factors need to be one-dimensional.

As an explicit example, let $L(\lambda),\lambda=\alpha/2$ be the 2-dimensional module for $U_{q}(\mathfrak{sl}_{2})$ with basis $x_{0}:=v_{\lambda},~{}x_{1}:=v_{\lambda-\alpha}$. Then the elements of the Borel subalgebra $B_{\lambda,\lambda^{\prime}}$ act as follows:

[TABLE]

Hence $L(\lambda)$ restricted to $B_{\lambda,\lambda^{\prime}}$ has a one-dimensional submodule $\langle x_{0}+\lambda(1-q^{-2})x_{1}\rangle$, where the elements $EK^{-1}+\lambda K^{-1},F+\lambda^{\prime}K^{-1}$ act with eigenvalues $\lambda q,\lambda^{\prime}q^{-1}$, and a one-dimensional quotient $L(\lambda)/(x_{0}+\lambda(1-q^{-2})x_{1})$ with eigenvalues $\lambda q^{-1},\lambda^{\prime}q$ .

3.3. Example: Homogeneous Borel subalgebras

We want to prove that all homogeneous Borel subalgebras of $U_{q}(\mathfrak{g})$ are standard Borel subalgebras i.e. reflections via $T_{w}$ of $U^{\leq 0}$. We start by proving that they are basic:

Lemma 3.10**.**

For any Weyl group element $w\in W$ consider the right coideal subalgebra $U^{-}[w]$ and the restriction of the counit $\epsilon:U\to\Bbbk$. Then any finite-dimensional irreducible representation $V$ on which all elements in $U^{-}[w]\cap\ker(\epsilon)$ act nilpotently is one-dimensional and hence the trivial representation $\Bbbk_{\epsilon}$.

Proof.

From Chapter 6 in [HS09] we know, that $U^{-}[w]$ is isomorphic to a smash product $k[F_{\alpha}]\#T_{\alpha}(U^{-}[s_{\alpha}w])$ with $s_{\alpha}<w$. We prove our claim by induction on the length $\ell(w)$:

For $w=1$ the claim is certainly true. Let the assertion be proven for $v\in W$ and consider the case $w:=s_{\alpha}v$ with $\ell(w)=\ell(v)+1$. Assume a representation $V$ on which $U^{-}[w]\cap\ker(\epsilon)$ acts nilpotently. Restricting to the subalgebra $T_{\alpha}(U^{-}[v])\subset U^{-}[w]$ induction provides that all composition factors are one-dimensional $\Bbbk_{\epsilon}$. In particular there is a common trivial eigenvector of this subalgebra $h.v=\epsilon(h)v$. Let $V_{0}\subset V$ be the nontrivial subspace

[TABLE]

This is clearly a $T_{\alpha}(U^{-}[v])$-subrepresentation. Moreover we know that $\mathrm{ad}_{F_{\alpha}}$ acts on $T_{\alpha}(U^{-}[v])$ in the smash product, so for any $X_{\mu}\in T_{\alpha}(U^{-}[v])_{\mu}$

[TABLE]

It follows that $F_{\alpha}$ preserves $V_{0}$ and because $V$ was assumed irreducible we get $V=V_{0}$. As $F_{\alpha}$ acts on $V$ nilpotently it has an eigenvector $v^{\prime}$ with eigenvalue [math], which is then a one-dimensional trivial representation of $U^{-}[w]$. ∎

In particular we know that all elements $F_{\alpha}\in U_{q}(\mathfrak{g})$ act nilpotently on $V$ (see [Jan96] Proposition 5.1). So the previous lemma shows that all irreducible subquotients of $U_{q}(\mathfrak{g})$-modules are trivial one-dimensional. For an arbitrary representation of $U^{-}[w]$ this is certainly not true, as already the example $U^{-}[s_{\alpha}]=\Bbbk[F_{\alpha}]$ shows, but it is to expect that $U^{-}[w]$ is still basic. For us it is however sufficient to show:

Lemma 3.11**.**

Given a Weyl group element $w\in W$, $L$ a subgroup of $\Lambda$ and the corresponding right coideal subalgebra $C=\Bbbk[L]U^{-}[w]$ with the property, that for all $\mu\in\Phi^{+}(w)$ there exists a $\nu\in L$ such that $(\mu,\nu)\neq 0$, then $C$ is basic.

In particular $U^{0}U^{-}[w]$ is basic, as well as the right coideal subalgebra $U^{0}U^{+}[w]$.

Proof.

The proof is similar to the proof of the Lemma 3.10: We show, that each finite dimensional irreducible representation is one-dimensional and that $U^{-}[w]$ acts on all one-dimensional representations by $\epsilon$.

As $\Bbbk[L]$ is abelian, the claim holds for $w=1$. Again we consider inductively the restriction of a $\Bbbk[L]U^{-}[w]$-module to the subalgebra $T_{s_{\alpha}}(U^{-}[v])$ where by induction we find a common eigenvector $v$ with $E_{\beta}.v=0$ and $K_{\beta}.v=\chi(K_{\beta})v$ for some character $\chi:L\to\Bbbk^{\times}$ extended trivially to $\Bbbk[L]T_{s_{\alpha}}(U^{-}[v])$. Again we consider the nontrivial subspace

[TABLE]

and again the $\mathrm{ad}_{F_{\alpha}}$-stability of $T_{s_{\alpha}}(U^{-}[v])$ in the smash product, and the assumption that $T_{s_{\alpha}}(U^{-}[v])$ acts trivially implies that $V_{0}$ is a $U^{-}[w]$-submodule.

Let $v^{\prime}$ be an eigenvector of $F_{\alpha}$, then the new issue in this proof is whether the eigenvalue $\lambda$ may be nontrivial. But using the assumed element $K_{\nu}$ with $(\alpha,\nu)\neq 0$ with commutator relation $K_{\nu}F_{\alpha}=q^{-(\alpha,\nu)}F_{\alpha}K_{\nu}$ with $q$ not a root of unity implies then an infinite family of eigenvectors with distinct eigenvalues $q^{-n(\alpha,\nu)}$, which is a contradiction. Hence $F_{\alpha}$ acts on $V_{0}$ on the eigenvector by [math] and we have again found a one-dimensional subrepresentation with trivial action of $U^{-}[w]$ as asserted. ∎

Theorem 3.12**.**

The basic right coideal subalgebra $U^{0}U^{-}[w]$ is maximal with this property and hence a Borel subalgebra. We call it the standard Borel subalgebra.

Conversely each homogeneous Borel subalgebra $B$ is as algebra isomorphic to the standard Borel subalgebra via some $T_{w}$. Explicitly $B=U^{+}[w]U^{0}U^{-}[w^{-1}w_{0}]$.

Proof.

Since every right coideal subalgebra $B\supset U^{-}[w]U^{0}$ is by definition homogeneous it is sufficient for both assertions to prove that any homogeneous basic right coideal subalgebra is contained in some $B=T_{w}(U^{-}U^{0})$.

By [HK11a] Prop. 2.1 every homogeneous right coideal subalgebra $B$ is triangular and together with the classification of homogeneous right coideal subalgebras of $U_{q}^{\pm}$ in [HS09] Thm 7.3 they are of the form:

[TABLE]

We are finished if we can prove from $B$ being basic that $\Phi^{+}(w)\cap\Phi^{+}(v)=\{\}$. So assume to the contrary that there is $\mu\in\Phi^{+}(w)\cap\Phi^{+}(v)$, so the root vectors $E_{\mu}K_{\mu}^{-1},F_{\mu}\in B$ and hence $K_{\mu}^{-2}\in B$. Let $\lambda$ be a dominant integral weight with $(\lambda,\mu)\neq 0$, then there exists in the $U_{q}(\mathfrak{g})$-module $L(\lambda)$ a $K_{\mu}^{-2}$-eigenvector with eigenvalue $\neq 1$, say the highest weight vector. But $B$ contains a copy of $U_{q}(\mathfrak{sl}_{2})$ and all one-dimensional representations of this require $K_{\mu}^{-2}$ to act $=1$. Thus the restriction of $L(\lambda)$ to $B$ has to have some higher-dimensional irreducible composition factor (note that the eigenvalue argument does not require semisimplicity). Hence $B$ is in this case not basic. ∎

4. Induction of one-dimensional characters

4.1. Definition and first properties

One of the reasons Borel subalgebras of a Lie algebra or quantum group are interesting, is because they can be used to construct induced modules and a category $\mathcal{O}$. An interesting implication of defining and classifying unfamiliar Borel subalgebras $B$ is to study the respective induced representations from $B$ to $H=U_{q}(\mathfrak{g})$ of any one-dimensional $B$-module $\Bbbk_{\chi}$:

[TABLE]

As defining property for Borel algebras $B$ we chose to generalize the Lie algebra term “maximal solvable” by “maximal with the property basic” i.e. all irreducible finite-dimensional representations are one-dimensional. This matches the upcoming purpose, since we can again use solely one-dimensional characters of $B$. For a right coideal subalgebra $B$ the category $\mathrm{Rep}(B)$ is not a tensor category, but a right $\mathrm{Rep}(H)$-module category via $M\otimes_{\Bbbk}V$ with $B$-action given by $b^{(1)}\otimes b^{(2)}\in C\otimes H$.

Lemma 4.1**.**

For a right coideal subalgebra of a Hopf algebra $B\subset H$, the induction functor ${\mathrm{Rep}(B)\to\mathrm{Rep}(H)}$ and the restriction functor $\mathrm{Rep}(H)\to\mathrm{Rep}(B)$ are both morphisms of right $\mathrm{Rep}(H)$-module categories.

Proof.

Restriction is a morphism of module categories with trivial structure map ${\mathrm{Res}(V\otimes W)=\mathrm{Res}(V)\otimes W}$ due to the right coideal subalgebra property. Similarly, Induction is a morphism of module categories with structure map given by

[TABLE]

[TABLE]

which is easily checked to be an $H$-module morphism. ∎

Remark 4.2**.**

In our examples $H$ is free as a $B$ module, which is very helpful in constructing the induced representations. This does not follow directly from the celebrated result of Skryabin [Skry06] of freeness over coideal subalgebras, since $H$ is infinite-dimensional. It would be helpful to have a version which applies in our case. We feel that $B$ being free over $U^{0}$ is the decisive property.

Our main interest is how induced representations of Borel subalgebras decompose as $U_{q}(\mathfrak{g})$-modules. For a first result, we use part of our defining property:

Lemma 4.3**.**

Let $H$ be a Hopf algebra and $B\subset H$ a basic right coideal subalgebra. Then any finite-dimensional irreducible $H$-module $V$ is a quotient of an induced module $H\otimes_{B}\Bbbk_{\chi}$ for some character $\chi$.

Proof.

Consider the restriction of the finite-dimensional module $V$ to $B$. Since $B$ is basic, the composition series of $V$ consists of one-dimensional $B$-modules and in particular there is a nontrivial $B$-module monomorphism $\Bbbk_{\chi}\to V$ for a suitable $\chi$. This induces up to a nonzero $H$-module morphism $H\otimes_{B}\Bbbk_{\chi}\to V$. For $V$ irreducible this morphism has to be surjective. ∎

Example 4.4**.**

Let $B=U^{\leq 0}$ be the standard Borel subalgebra of $U_{q}(\mathfrak{g})$, then every one-dimensional character $\chi:B\to\Bbbk$ is zero on $U^{-}$ and thus comes from some group character $\chi:\Lambda\to\Bbbk^{\times}$. The induced module $U\otimes_{B}\Bbbk_{\chi}$ is isomorphic to $U^{+}$ as a $U$-module. It is a highest-weight module generated by the vector $v$ with $K_{\mu}.v=\chi(K_{\mu})v$ and $E_{\alpha}.v=0$.

We shall see that also for our new Borel subalgebras $B$ the induced modules for generic $\chi$ are irreducible, while special values of $\chi$ have as quotient each of the irreducible finite-dimensional $U_{q}(\mathfrak{g})$-modules $L(\lambda)$. However the action of $U^{0}$ will not be diagonalizable any more.

Problem 4.5**.**

Is it true that any induced module from a Borel subalgebra have a unique irreducible quotient? It is to expect that maximality enters here.

Problem 4.6**.**

Can one use the graded algebra in the upcoming Conjecture A to understand these modules and determine their decomposition behaviour of the induced modules? We would expect that $\mathrm{gr}(B)^{0}$ plays a similar role as $U^{0}$ in the standard case, so the graded modules $\mathrm{gr}(V)$ should be $\mathrm{gr}(B)^{0}$-diagonalized.

Problem 4.7**.**

Can one define a category $\mathcal{O}$ with nice properties? In particular it should be an abelian category, closed under submodules and quotients, with enough projectives and injectives.

4.2. Example $A_{1}$

We now want to construct and decompose all induced representations in the case $\mathfrak{sl}_{2}$. From Section 3.2 we know all Borel algebras of $U_{q}(\mathfrak{g})$ up to reflection:

In the familiar case $B_{std}=U^{\leq 0}=\langle K,F\rangle$ all one-dimensional representations are of the form $\chi(F)=0$ and $\chi(K)$ arbitrary. Then the induced module is

[TABLE]

with highest weight vector $K.1_{\chi}=\chi(K)1_{\chi}$. All of these modules have a diagonal action of $K$. For certain integral choices $V(B_{std},\chi)$ has a finite-dimensional irreducible quotient and all finite-dimensional irreducible $U_{q}(\mathfrak{g})$-representations arise this way.

The novel case in Example 3.6 is $B_{\lambda,\lambda^{\prime}}:=\langle\bar{E},\bar{F}\rangle$ with $\bar{E}:=EK^{-1}+\lambda K^{-1},\bar{F}=F+\lambda^{\prime}K^{-1}$ and arbitrary scalars $\lambda\lambda^{\prime}=\frac{q^{2}}{(1-q^{2})(q-q^{-1})}$, an algebra isomorphic to the quantized Weyl algebra. As we showed, all finite-dimensional irreducible representations factorize over the commutative quotient $\langle\bar{E},\bar{F}\rangle/(\bar{E}\bar{F}-\frac{q^{2}}{(q-q^{-1})(1-q^{2})})$. Hence all one-dimensional representations are of the form $\chi(\bar{E})=e$, $\chi(\bar{F})=f$ with $ef=\frac{q^{2}}{(q-q^{-1})(1-q^{2})}=\lambda\lambda^{\prime}$. Using the PBW basis we easily get

[TABLE]

and we calculate the action to be

[TABLE]

or in matrices

[TABLE]

In particular, the action of $K$ is not diagonalizable. Moreover the action of $K$ shows that no $V(B_{\lambda,\lambda^{\prime}},\chi)$ has finite-dimensional proper submodules.

We can determine all submodules with a trick that does not seem to easily generalize beyond rank one:

Lemma 4.8**.**

The induced $U_{q}(\mathfrak{sl}_{2})$-module with respect to a Borel algebra $B_{\lambda,\lambda^{\prime}}$

[TABLE]

has a nontrivial submodule $V^{\prime}$ iff for some $n\in\mathbb{N}_{0}$

[TABLE]

It is cofinite of codimension $[V:V^{\prime}]=n+1$.

Proof.

A $U_{q}(\mathfrak{sl}_{2})$-submodule $W\subset V(B_{\lambda,\lambda^{\prime}},\chi)\cong\Bbbk[K,K^{-1}]$ is in particular a $\Bbbk[K,K^{-1}]$-submodules under left-multiplication i.e. an ideal (instead of weight-spaces). Since this is a principal ideal ring, there exists a Laurent-Polynomial $P(X)=\sum_{n\in\mathbb{Z}}^{finite}c_{n}X^{n}$ with $W=(P),X=K$. This is a submodule iff $E.P$ and $F.P$ are multiples of $P$. We calculate explicitly:

[TABLE]

For degree reasons this can only give a multiple of $P$ if for some $a,b,c,d$:

[TABLE]

Either $P(X)$ is constant, then $W$ is the entire module, or the zeroes of $P$ have to lay in a chain $q^{-n}X_{0},q^{-n+2}X_{0},\ldots q^{n}X_{0}$ for some $n\in\mathbb{N}_{0}$, with $q^{n}X_{0}=\lambda^{\prime}/f$ and $q^{-n}X_{0}=\lambda/e$. The quotient module is thus the $(n+1)$-dimensional ring extension $\Bbbk[K,K^{-1}]/(P)$, where $K$ acts again by left-multiplication. ∎

We show a different and quite general idea to detect the finite-dimensional quotient modules (i.e. the cofinite submodules) and circumvents the problem of non-diagonalizable $K$-action: To find finite-dimensional quotient modules we need to find elements in $\mathrm{Hom}_{U_{q}(\mathfrak{g})}(U_{q}(\mathfrak{g})\otimes_{B}\Bbbk_{\chi})$ and this can be done by decomposing $\mathrm{Hom}_{\Bbbk}(U_{q}(\mathfrak{g}),\Bbbk)$ into irreducible $L(\lambda)$. This is now possible since $\mathrm{Hom}_{\Bbbk}(U^{0},\Bbbk)$ is diagonalizable under left-multiplication - morally because it is the algebraic closure. The following Lemma verifies explicitly the existence assertion in Lemma 4.3.

Lemma 4.9**.**

The induced $U_{q}(\mathfrak{sl}_{2})$-module with respect to a Borel algebra $B_{\lambda,\lambda^{\prime}}$

[TABLE]

has a finite-dimensional quotient the irreducible module $L(\epsilon,n)$ of dimension $n+1$ and sign $\epsilon=\pm 1$ for precisely one choice of $\chi$, namely

[TABLE]

Proof.

Let $\phi:V(B_{\lambda,\lambda^{\prime}},\chi)\to L(\epsilon,n)$ be a (nonzero) module homomorphism to the irreducible highest weight module with highest weight vector $v_{0}$ for highest weight $Kv=q^{n}v$. It has a basis $v_{0},\ldots v_{n}$ with $Kv_{i}=\epsilon q^{n-2i}v_{i}$. Denote the coefficients of $\phi$ in this basis by $\phi_{k},\;k=0\ldots n$,. Then by definition

[TABLE]

Thus $\phi$ is fixed by its image $\phi_{k}(1_{\chi})$ via

[TABLE]

and since the map should be surjective we need all $\phi_{k}(1_{\chi})\neq 0$ for $k=0\ldots n$.

On the other hand the action of $F$ and $E$ demands:

[TABLE]

and the boundary conditions of the first resp. second equation for $k=-1$ resp. $k=n$ that the right-hand side has to be zero

[TABLE]

which is the asserted condition. One could also derive direct formulae for $\phi_{k}(K^{i}1_{\chi})$ and in this case it is easy to see that $\phi$ above is indeed a module homomorphism.

∎

5. The structure of the graded algebra of a right

coideal subalgebra

5.1. Conjecture A

By Theorem 2.5, right coideal subalgebras of $U^{-}_{q}(\mathfrak{g})$ are character shifts $\Bbbk[L]U^{-}[w]_{\phi}$ with $L\subset\Lambda$ orthogonal to $\mathrm{supp}(\phi)$. These character-shifts are not $\Lambda$-graded. An important structural insight into these algebras, and subsequently right coideal subalgebras of $U_{q}(\mathfrak{g})$ and there representation theory, would be the knowledge of the respective graded algebras. Take the $\mathbb{Z}$-grading of $U_{q}(\mathfrak{g})$ where $\deg(E_{\alpha_{i}})=1$, $\deg(K_{\alpha_{i}})=0$, $\deg(F_{\alpha_{i}})=-1$. then we conjecture:

Conjecture A**.**

Fix some $w\in W$ and character $\phi$ and consider the canonical map $f:\mathrm{gr}(U^{-}[w]_{\phi})\to U^{\leq 0}$ sending an element to the summands of leading degree. Then the image $D$ of $f$ is a $\mathbb{Z}$-graded right coideal subalgebra of $U^{\leq 0}$ and has the following properties:

• •

$D^{0}$* is the semigroupring of $G(D^{0})=\{K_{\mu}^{-1}\mid\mu\in\mathrm{supp}(\phi)\}$. We denote by $\tilde{G}$ its quotient group or localization.*

• •

Then for the explicit element $w^{\prime}=(\prod_{\beta\in\mathrm{supp}(\phi)}s_{\beta})w$ we have

[TABLE]

Morally*, this assertion means $\mathrm{gr}(U^{-}[w]_{\phi})=U^{-}[w^{\prime}]$ up to localization, and the main part of the assertion is the explicit formula of $w^{\prime}$.*

The first step of the proof of Conjecture A is to observe that $D$ is a right coideal subalgebra of $U^{-}_{q}(\mathfrak{g})$. If $D^{0}=\Bbbk[\mathbb{Z}^{k}]$ is a full group ring, then Theorem 2.3 asserts that that $D\cong U^{-}[w^{\prime}]$ for some $w^{\prime}\in W$. However, $D^{0}$ need not be a group so there are many possible (closely related) $D$ generated by linear combinations of character-shifted $E_{\alpha},F_{\alpha}$ with different $K$-prefactors, see Example 5.5. To circumvent these problems, we localize $G$ to $\tilde{G}$, then Theorem 2.3 asserts $\Bbbk[\tilde{G}(D^{0})]D=\Bbbk[\tilde{G}(D^{0})]U^{-}[w^{\prime}]$. Our task is to prove the asserted formula for $w^{\prime}$, which captures the information which $E_{\alpha},F_{\alpha}$ appear in both $D$ and its localization.

We prove Conjecture A for type $A_{n},B_{n},C_{n},D_{n}$ in this section. We first give some examples and formulate the consequence for triangular coideal subalgebras.

Example 5.1**.**

Clearly for $\phi=0$ the right coideal subalgebra $U^{-}[w]_{\phi}$ is already graded and $w^{\prime}=w$.

Example 5.2**.**

Take our example $B_{\lambda,\lambda^{\prime}}$ with negative part $U^{-}[s_{\alpha}]_{\phi_{-}}=\langle F+\lambda^{\prime}K^{-1}\rangle$ for $\phi_{-}(F)=\lambda^{\prime}\neq 0$. Then the graded algebra is as conjectured with $w=s_{\alpha_{1}},w^{\prime}=1$:

[TABLE]

Altogether, the graded algebra of $B_{\lambda,\lambda^{\prime}}$ is the positive Borel part, up to localization:

[TABLE]

Example 5.3**.**

Let $\mathfrak{g}=\mathfrak{sl}_{3},w=w_{0}=s_{1}s_{2}s_{1}$ and $\chi(F_{1})=\lambda^{\prime},\mathrm{supp}(\chi)=\{\alpha_{1}\}$. Then $U^{-}[w]_{\chi}=\langle F_{1}+\lambda^{\prime}K_{1}^{-1},F_{2}\rangle$. The leading degree terms of the character-shift $\bar{F}_{1}=F_{1}+\lambda^{\prime}K_{1}^{-1}$ is $\lambda^{\prime}K_{1}^{-1}$, and $\bar{F}_{2}=F_{2}$. Typically, character-shifts of elements of $\Lambda$-degree $-(\alpha_{1}+\alpha_{2})$ have leading degree terms in $\Lambda$-degree $-\alpha_{1}$:

[TABLE]

However, in a suitable linear combination of these two elements, the leading degree terms of the character-shifts cancel and we again land in $\Lambda$-degree $-(\alpha_{1}+\alpha_{2})$:

[TABLE]

Thus the graded algebra is as conjectured, with $w^{\prime}=s_{1}w=s_{1}s_{2}$:

[TABLE]

We remark that the definition of a root vector depends on the choice of a reduced expression for $w$, while the linear combination we found above does not, so it can be a root vector or not.

In contrast, for a right coideal subalgebra in the positive part $U^{0}S(U^{+})$ the graded right coideal subalgebra has the same degrees, i.e. by choice of our grading the leading terms of the character-shifted $\bar{E}_{\alpha}$ are simply the $E_{\alpha}$ themselves.

[TABLE]

The two statements can be easily combined:

Corollary 5.4**.**

Assuming Conjecture A holds for $U_{q}[w_{-}]_{\phi_{-}}$ i.e. for $D:=\mathrm{gr}(U_{q}[w_{-}]_{\phi_{-}})$ holds after completing the semigroup $G(D^{0})$ to the quotient group $\tilde{G}(D^{0})$:

[TABLE]

Then for any triangular right coideal subalgebra the map sending elements to their leading degree terms gives an isomorphism of right coideal subalgebras

[TABLE]

As the representation theory of the graded algebra is well understood, we would ultimately hope that this puts explicit and sharp condition when a choice of $(w_{+},\phi_{+},w_{-},\phi_{-})$ produces a triangular Borel algebras and thus a classification. This is stated in the next section as Conjecture B and we will prove one direction.

5.2. Localization

Let in this section $A$ be a right coideal subalgebra. A technical difficulty, which we address in this section is that in general $A^{\prime}=\mathrm{gr}(A)$ has in general $A^{\prime}=\Bbbk[G]$ for a semigroup $G\subset\mathbb{Z}^{k}$. Such a $\mathbb{Z}$-graded right coideal subalgebra may not even be $\mathbb{N}^{k}$-graded. Differently spoken, it does not have to be a free module over the semigroup $G$.

Example 5.5**.**

Consider for example the subalgebra in $U_{q}(\mathfrak{sl}_{3})^{-}$ generated as:

[TABLE]

*It is closed under comultiplication and thus a right coideal subalgebra.

Multiplication with $K_{1}^{-1}$ or $K_{3}^{-1}$ from either side commute differently with $F_{1},F_{3}$, which shows that the following are elements in $C$

[TABLE]

but neither $F_{1},F_{3}$ are in $C$.

These $\mathbb{Z}$-graded right coideal subalgebras $A^{\prime}$ are not covered by the classification Theorem 2.3. We shall not try to classify $A^{\prime}$ directly, but study its localization:

Proposition 5.6**.**

Let $A^{\prime}\subset U^{-}$ be a $\mathbb{Z}$-graded right coideal subalgebra, and define the semigroup $G$ by $A^{\prime 0}=A^{\prime}\cap U^{0}=\Bbbk[G]$, then $\Bbbk{G}A^{\prime}=A^{\prime}\Bbbk{G}$ and thus we can define a localization in the noncommutative setting [Sten75] Prop 1.4:

Let $\tilde{G}\subset\mathbb{Z}^{k}$ be the quotient group (localization) of $G$. Then $\tilde{A^{\prime}}:=\Bbbk[\tilde{G}]A^{\prime}=A^{\prime}\Bbbk[\tilde{G}]$ is again a right coideal subalgebra and there exists a $w^{\prime}\in W$ with

[TABLE]

Proof.

Since $A^{\prime}$ is an algebra we have $KA^{\prime}=A^{\prime}K$ for $K\in G$, and by invertibility of $K$ we have $A^{\prime}K^{-1}=K^{-1}A^{\prime}K$, which implies the first claim. Now, $\tilde{A^{\prime}}$ is $\mathbb{Z}$-graded, so $\tilde{A^{\prime}}=\tilde{A^{\prime}}^{\leq}\tilde{A^{\prime}}^{\geq}$, and after localization both factors fulfills the conditions of Theorem 2.5 and are thus character-shifts. But since they are $\mathbb{Z}$-graded, the character-shift has to be trivial and hence $\tilde{A^{\prime}}$ is as asserted, in particular $\mathbb{Z}^{n}$-graded. ∎

Proposition 5.7**.**

Let $G$ be a semigroup with $\tilde{G}$ its quotient group. Let $\tilde{V}$ be a $\tilde{G}$-module and $V$ be a $G$-submodule, such that $\tilde{G}V=\tilde{V}$. Then for any $v\in\tilde{V}$ there exists a $K\in G$ such that $Kv\in V$.

Proof.

Since we assumed $\tilde{G}V=\tilde{V}$, there exist finitely many elements $K^{(i)}\in\tilde{G}$ and vectors $v^{(i)}\in V$ such that

[TABLE]

Now by definition of $\tilde{G}$ there exists for each element $K^{(i)}$ an element $K_{(i)}\in G$ with $K_{(i)}K^{(i)}\in G$. Taking some $K\in G$ in $\bigcap_{i}K_{(i)}G$ (for example their product) proves the assertion. ∎

Applying this to $\Bbbk[\tilde{G}]A^{\prime}=\Bbbk[\tilde{G}]U^{-}[w^{\prime}]$ we get the following easy consequence, which will also be useful later-on:

Corollary 5.8**.**

For any root $\mu\in\Phi(w^{\prime})$ and a reduced expression for $w^{\prime}$, there exists a $K\in G$, such that $KF_{\mu}\in A^{\prime}$.

Next, we define a suitable growth-condition that replaces the length $\ell(w^{\prime})$ respectively the number of PBW generators for arbitrary non-homogeneous $A^{\prime}$:

Definition 5.9**.**

Let $V$ be an $\mathbb{N}$-graded vector space with finite-dimensional homogeneous components $V_{n}$. Let $\mathcal{H}(V,z)=\sum_{n\geq 0}\dim(V_{n})z^{n}$ be the Hilbert series, then we define $\mathrm{growth}(V)=a$ such that

[TABLE]

otherwise we say the growth is undefined.

The definition is made in a way, such that $V^{sub}\subset V\subset V^{sup}$ with well-defined $\mathrm{growth}(V^{sub})=\mathrm{growth}(V^{sup})=:a$ implies an intermediate well-defined $\mathrm{growth}(V)=a$. Note that $\mathcal{H}(V,z)$ needs not be a rational function, even if $\mathcal{H}(V^{sub},z),\mathcal{H}(V^{sup},z)$ is; note further that the limits need not coincide.

Example 5.10**.**

If $V$ is an graded algebra with a basis of sorted monomials in generators $x_{1},\ldots x_{\ell}$ of degree $d_{1},\ldots d_{\ell}\geq 1$, then $\mathrm{growth}(V)=\ell$, since explicitly

[TABLE]

When we apply this notion and result to our $A,A^{\prime}\subset U^{-}$, we need to consider a modified grading, which has finite-dimensional homogeneous components, but equal behaviour with respect to character shifts: Define the modified degree $\mathrm{mdeg}(-)=\deg(-)+\mathrm{sdeg}(-)$ by adding a second degree $\mathrm{sdeg}(F_{i})=\mathrm{sdeg}(K_{i}^{\pm 1})=-1$, so $\mathrm{mdeg}(F_{i})=-2$ and $\mathrm{mdeg}(K_{i}^{\pm 1})=-1$. Note that the algebra is not graded any more. Clearly the homogeneous components of $\mathrm{mdeg}(-)$ on $U^{-}$ are finite-dimensional. Since characters shifts are graded with respect to $\mathrm{sdeg}(-)$, this new definition does not change the vector space $\mathrm{gr}(A)$:

[TABLE]

Example 5.11**.**

Let $A=U^{-}[w]$ with the modified grading, then ${\mathrm{growth}(A)=\ell(w)}$. More generally, let $G$ be a semigroup with $\tilde{G}=\mathbb{Z}^{k}$ and take $A^{\prime}=\Bbbk[G]U^{-}[w^{\prime}]$, then $\mathrm{growth}_{\Bbbk}(A)=\mathrm{rank}{\tilde{G}}+\ell(w^{\prime})$.

We now need to compare the growth of $A=U^{-}[w]_{\phi}$, or equivalently $A^{\prime}=\mathrm{gr}(A)$, to its localization $\Bbbk[\tilde{G}]A^{\prime}=\Bbbk[\tilde{G}]U^{-}[w^{\prime}]$:

Lemma 5.12**.**

Let $V$ be a $\mathbb{N}$-graded module over a semigroup $G\subset\mathbb{Z}^{k}$. Then we have for the localization We have for the localization $\mathrm{growth}(V)=\mathrm{growth}(\Bbbk[\tilde{G}]V)$.

Proof.

Let $\tilde{V}=\tilde{G}V$, which is a free $\Bbbk[\tilde{G}]$-module. Let $I$ be a $\tilde{G}$-basis of $\tilde{V}$, then by Proposition 5.7 there are elements $K_{(i)}\in G$ with $K_{(i)}v_{i}\in V$ and we define its span to be $V^{sub}\subset V$. On the other hand we may consider $V^{sup}=\tilde{V}\supset V$.

Then we have an inequality by definition

[TABLE]

but in our case $\mathrm{growth(V^{sub})}=\mathrm{growth(V^{sup})}$, which proves the assertion ∎

Corollary 5.13**.**

Let now again $A:=U^{-}[w]_{\phi}$ and $A^{\prime}:=\mathrm{gr}(A)$ and the localization $\Bbbk[\tilde{G}]A^{\prime}=\Bbbk[\tilde{G}]U^{-}[w^{\prime}]$. Then comparing the growth gives

[TABLE]

5.3. Induction step

Let $w\in W$ and $\phi$ be a character on $U^{-}[w]$ with support $\mathrm{supp}(\phi)\subset\Phi^{+}(w)$, a set of orthogonal simple roots. We continue to study the character-shifted coideal subalgebra, its graded algebra and its localization

[TABLE]

for some $w^{\prime}\in W$, whose explicit description is the goal of Conjecture A. We attempt an induction on the length $\ell(w)$.

Lemma 5.14**.**

Assume Conjecture A holds for all $u\in W$ with $\ell(u)<\ell(w)$ and all characters. Assume $w=us_{\alpha_{i}}$ with $\ell(w)=\ell(u)+1$ for some $\alpha_{i}$. Then one of the following happens:

• •

If $u(\alpha_{i})\in\mathrm{supp}(\phi)$ then Theorem Conjecture A holds for $w$ and $w^{\prime}=u^{\prime}$, which is the $w^{\prime}$ asserted in this case.

• •

If $u(\alpha_{i})\in\mathrm{supp}(\phi)$ then there exists a simple root $\alpha_{j}$ such that $w^{\prime}:=u^{\prime}s_{\alpha_{j}}$ has length $\ell(u^{\prime})+1$ and

[TABLE]

Conjecture A holds for $w$ iff $\alpha_{j}=\alpha_{i}$.

Proof.

By [HK11b] p. 13 for $u<w$ the restriction $\phi_{u}$ of $\phi$ to $U^{-}[u]\subset U^{-}[w]$ is again a character with $\mathrm{supp}(\phi_{u})\subseteq\mathrm{supp}(\phi)$. The inductive assumption that Theorem Conjecture A holds for $U^{-}[u]_{\phi_{u}}$ means that we have

[TABLE]

where $u^{\prime}$ has the asserted form $u^{\prime}:=(\prod_{\beta\in\mathrm{supp}(\phi_{u})}s_{\beta})u$, and where $\tilde{G}_{u}$ is the quotient group of the semigroup $G_{u}$ generated by $\mathrm{supp}(\phi_{u})$, with $A^{\prime 0}_{u}=\Bbbk[G_{u}]$.

Now for the new element $w\in W$ with $\Phi^{+}(w)=\Phi^{+}(u)\cup\{u(\alpha_{i})\}$ a new character-shifted root vector $\bar{F}_{u(\alpha_{i})}$ appears in the PBW-basis

[TABLE]

We now apply Corollary 5.13 which asserts

[TABLE]

[TABLE]

We have $\ell(w)=\ell(u)+1$. Moreover by inclusion $A^{\prime 0}\supseteq A^{\prime 0}_{u}$ and $\Phi^{+}(w^{\prime})\supseteq\Phi(u^{\prime})$ we have inequalities $\mathrm{rank}(\tilde{G})\geq\mathrm{rank}(\tilde{G}_{u})$ and $\ell(w^{\prime})\geq\ell(u^{\prime})$. Hence we have two possible cases:

1. i)

$\mathrm{rank}(\tilde{G})=\mathrm{rank}(\tilde{G}_{u})+1$ and $\ell(w^{\prime})=\ell(u^{\prime})$. The latter implies $w^{\prime}=u^{\prime}$, the former implies $\mathrm{rank}(\tilde{G})\gneq\mathrm{rank}(\tilde{G}_{u})$. Hence the leading term of $\bar{F}_{u(\alpha_{i})}$ must be the new element $K^{-1}_{\mu}\in A^{\prime 0}$. By orthogonality of $\mathrm{supp}(\phi)$, we find that $\mu=u(\alpha_{i})$ is a new simple root $\alpha_{n}$ in $\mathrm{supp}(\phi)=\{\mu\}\cup\mathrm{supp}(\phi_{u})$. In particular $w=us_{\alpha_{i}}=s_{\alpha_{n}}u$. This proves that $w^{\prime}=u^{\prime}$ is as asserted by Conjecture A as

[TABLE] 2. ii)

$\mathrm{rank}(\tilde{G})=\mathrm{rank}(\tilde{G}_{u})$ and $\ell(w^{\prime})=\ell(u^{\prime})+1$. The former implies $\tilde{G}=\tilde{G}_{u}$ and $\mathrm{supp}(\phi)=\mathrm{supp}(\phi_{u})$. The second implies together with $\Phi^{+}(w^{\prime})\supset\Phi^{+}(u^{\prime})$ that there exists a simple root $\alpha_{j}$ with $w^{\prime}=u^{\prime}s_{\alpha_{j}}$. Since the support is unchanged, the assertion of Conjecture A would be $\alpha_{j}=\alpha_{i}$.

∎

The second case is what we need to study in what follows. We remark that the new element in the graded algebra in degree $u^{\prime}(\alpha_{j})\in\Phi^{+}(w^{\prime})$ does not have to be directly the leading term of the character-shift $\overline{F}_{u(\alpha_{i})}$. We illustrate this in an explicit example:

Example 5.15**.**

Let $\mathfrak{g}=A_{3}$ and $u=s_{3}s_{1}s_{2}$, then

[TABLE]

The algebra $U^{-}[s_{3}s_{1}s_{2}]$ has a PBW basis in the generators

[TABLE]

In accordance with the previous theorem, the character-shifted root vectors

[TABLE]

have highest degrees $\{0,0,\beta_{2}\}$ and thus the graded algebra corresponds to the Weyl group element $u^{\prime}=s_{12}s_{23}u=s_{2}$ as conjectured:

[TABLE]

Now the critical induction step is to prolong this by $\alpha_{i}=\beta_{1}$ to $w=us_{1}$, then we have a new root vector in degree $u(\alpha_{i})=\beta_{23}$, to be precise:

[TABLE]

The leading degree term $F_{2}$ of $\bar{F}_{23}$ is already contained in $U^{-}[u^{\prime}]$, as leading degree term of $\bar{F}_{{123}}$. The new degree $u^{\prime}(\alpha_{j})$ in $U^{-}[w^{\prime}]\supset U^{-}[u^{\prime}]$ will be the leading degree of some suitable linear combination, and the main question is which will be the next-leading degree. Simply from degree considerations there are two reasonable possibilities (in the sense of criterion 1 below):

• •

$u(\alpha_{i})=u^{\prime}(\alpha_{j})=\beta_{23}$, contrary to Conjecture A.

Actually this seems the natural choice, because it is the next leading degree in $\bar{F}_{23}$.

• •

$u(\alpha_{i})+\beta_{12}-\beta_{23}=u^{\prime}(\alpha_{i})=\beta_{12}$, according to Conjecture A.

For this to happen, $F_{2},{F}_{23}$ have to cancel simultaneously in a suitable linear combination of $\bar{F}_{123},\bar{F}_{23}$, and then ${F}_{12}$ from $\bar{F}_{23}$ is the leading term.

Having explicit expressions for $\bar{F}_{123},\bar{F}_{23}$ we can check explicitly that the latter is the case, because the coefficients of $F_{2},{F}_{23}$ in $\bar{F}_{123},\bar{F}_{23}$ are proportional:

[TABLE]

We note that already the non-vanishing of the $F_{12}$-term would have sufficed in this situation, because neither $\beta_{12},\beta_{23}$ are larger then the other and we know from length that we only get a single new degree.

5.4. Three Criteria implying Conjecture A

Let $C=U^{-}[w]_{\phi}$ with $w=us_{\alpha_{i}},u(\alpha_{i})\neq\mathrm{supp}(\phi)$ and let $\mathrm{gr}(U^{-}[u]_{\phi})=U^{-}[u^{\prime}]$ for some $u^{\prime}$, then the second case in Lemma 5.14 states that $\mathrm{gr}(U^{-}[w]_{\phi})=U^{-}[w^{\prime}]$ for $w^{\prime}=u^{\prime}s_{\alpha_{j}}$. Conjecture A is concerned with the shape of $u^{\prime},w^{\prime}$, and it would hold inductively if $\alpha_{j}=\alpha_{i}$.

We now collect three criteria when this is the case. Each criterion by itself shows the induction step for certain cases $w=us_{\alpha_{i}}$. Besides they give the possibility to check it manually in a given case - this is in particular interesting for Criterion 3.

As first criterion we note the general relation between $\alpha_{i}$ and (the potentially different) $\alpha_{j}$, which allows us to classify critical induction steps up to reflection.

Lemma 5.16** (Criterion 1).**

1. a)

We have the necessary criterion $u(\alpha_{i})-u^{\prime}(\alpha_{j})\in\mathbb{Z}[\mathrm{supp}]$. 2. b)

Assume (contrary to Conjecture A) that $\alpha_{i}\neq\alpha_{j}$ for $u(\alpha_{i})-u^{\prime}(\alpha_{j})\in\mathbb{Z}[S]$ with $S\subset\mathrm{supp}$ a subset of $S$, which we choose minimal. Then $u(\alpha_{i}),u^{\prime}(\alpha_{j}),S$ are up to reflection in a parabolic subsystems $\Phi_{r}\subset\Phi$ of rank $3,4$ as follows:

[TABLE]

or $u(\alpha_{i})=u^{\prime}(\alpha_{j})$. If Conjecture A holds already for $u$, then we have in addition $u(\alpha_{i})-u(\alpha_{j})\in\mathbb{Z}[S]$, so also this tuple is in the list above.

Proof.

1. a)

The proof of Lemma 5.14 shows that $U^{-}[w]_{\phi}$ contains a new element $\bar{F}_{u(\alpha_{i})}$ and since $\phi$ is a character-shifts with respect to $\mathrm{supp}$, this algebra is still graded by cosets of $\mathbb{Z}[\mathrm{supp}]$. On the other hand the graded algebra $A^{\prime}$ contains a new element in degree $u^{\prime}(\alpha_{j})$. 2. b)

Any set of $k$ roots can be reflected to a parabolic subsystem of rank at most $k$. Applying this assertion to $S,\beta_{i}$ gives a subsystem $\Phi_{r}$ of rank $r=|S|+1$, which is moreover connected (otherwise the equation would hold even for a subset $S^{\prime}$ smaller $S$). Applying the assertion a second times to $S$ gives a subset of $|S|=r-1$ many orthogonal simple roots. Since $|S|>1$, this can only be true for $\Phi_{r}=A_{3},B_{3},C_{3}$ with $S$ reflected to the outmost simple roots $\alpha_{1},\alpha_{3}$, or for $\Phi_{r}=D_{4}$ with $S$ reflected to the three outmost roots $\alpha_{1},\alpha_{2},\alpha_{3}$. The result then follows from directly inspecting these root systems for solutions $\beta_{i}-\beta_{j}\in\mathbb{Z}[S]$.

∎

As a second criterion, we formulate the consequence of applying the induction step to two different presentations of $w$:

Definition 5.17**.**

A Weyl group element $w$ has a unique ending iff one of the following equivalent conditions applies:

• •

Any reduced expression of $w$ ends in the same letter $s_{\alpha_{i}}$.

• •

$\Phi^{+}(w^{-1})$* contains a unique simple root $\alpha_{i}$.*

• •

$w(\alpha_{k})>0$* for all simple roots $\alpha_{k}$ expect for one $\alpha_{i}$.*

Lemma 5.18** (Criterium 2).**

Assume that $w$ has not a unique ending, i.e. $w=u_{1}s_{\alpha_{i}}=u_{2}s_{\alpha_{l}}$ with $\alpha_{i}\neq\alpha_{l}$, and $u_{1}(\alpha_{i}),u_{2}(\alpha_{l})\not\in\mathrm{supp}$. Then if Conjecture A holds for $u_{1},u_{2}$, then $\alpha_{j}=\alpha_{i}$ and thus Conjecture A holds also for $w$.

Proof.

As is well known from [M64] Thm. 29, there is a series of braid group moves to transform the two presentations of $w$ into one another. In particular under our assumption there exists a reduced expression of $w$ which ends

[TABLE]

of length $2,3,4,6$ depending on $(\alpha_{l},\alpha_{i}),i\neq l$, such that a braid group move can be performed. Differently spoken

[TABLE]

[TABLE]

Assume that Conjecture A holds for the elements $u_{1},u_{2}$ of shorter length, then Lemma 5.14 asserts that there exists $\alpha_{j},\alpha_{k}$ with

[TABLE]

and in particular

[TABLE]

In case we have $\alpha_{i}\neq\alpha_{j}$, this would imply already $\Phi(u^{\prime}_{1})\subset\Phi(u^{\prime}_{2})$, which is a contradiction. ∎

As a third criterion we prove the assertion about $U^{-}[w]$ for $w=us_{\alpha_{i}}$ by direct calculation. In general, it is not easy to relate the coproduct of a reflected element to the coproduct of the original element, but we can obtain enough information under a multiplicity-zero condition. It enters through the following formulae:

Proposition 5.19**.**

Let $\alpha_{m}$ be a simple root and consider $T_{v^{-1}}^{-1}(F_{i})$ for any $v\in W,i\in I$, which is in degree $\nu:=v(\alpha_{i})$. Assume that $(\alpha_{m},\nu)=-1$ and that $\alpha_{m}$ has multiplicity zero in the degree of $v(\alpha_{i})$. Then

[TABLE]

If $(\alpha_{m},\nu)=-c$ a similar formula holds.

Proof.

To compute the character-shift $(T_{s_{m}}T_{v^{-1}}^{-1}(F_{i}))_{\phi_{m}}$ we need to relate coproduct and reflection, which is involved. From [Lusz93] Prop 37.3.2 we have the formula***We use the notation $T_{s_{m}},T_{s_{m}}^{-1}$ in [Jan96], which translates by the remark on p. 146 to Lusztig’s notation $T_{m,1}^{\prime\prime},T_{m,{-1}}^{\prime}$.

[TABLE]

where $\{l\}_{q^{m}}=[l]_{q_{m}}(q_{m}-q_{m}^{-1})^{l}$. We need an analogous formula for $T_{s_{m}}^{-1}$, by substituting $x=T_{s_{m}}^{-1}(y)$ and using $L^{\prime}L^{\prime\prime}=1$ from [Lusz93] Prop. 5.3.2

[TABLE]

If we apply the character $\phi_{m}\otimes 1$, then the only nonzero contributions are those tensor summands of $(T_{s_{m}}^{-1}\otimes T_{s_{m}}^{-1})\Delta(y)$ and hence $\Delta(y)$ where the left tensor factor has degree in $\mathbb{Z}\alpha_{m}$.

We now use our assumption that $y:=T_{v^{-1}}^{-1}(F_{i})$ in degree $v(\alpha_{i})$ does not contain a factor $E_{m}$. Thus in this case the only nonzero contribution comes from the summand $1\otimes T_{s_{m}}^{-1}T_{v^{-1}}^{-1}(F_{i})$ in $(T_{s_{m}}^{-1}\otimes T_{s_{m}}^{-1})\Delta(y)$, thus in this case

[TABLE]

By reasons of degree (or by the quantum Serre relation), all terms $s\neq 0,1$ vanish for $(\alpha_{m},\nu)=-1$. If $(\alpha_{m},\nu)=-c$, then similarly only the terms up to $s=c$ are nonzero.

The last equality can then by most easily calculated by hand, since by the assumption on the multiplicity of $\alpha_{m}$, we have that $T_{v}(F_{i})$ is a linear combinations of terms $F^{\prime}F_{k}F^{\prime\prime}$ where $F^{\prime},F^{\prime\prime}$ are products of $F_{l}$ with $(\alpha_{m},\alpha_{l})=0$. Then by the multiplicativity of $T_{s_{m}}^{-1}$ and the defining formulae [Jan96] Sec 8.14 (8’) we have

[TABLE]

and similarly by $[E_{m},F_{l}]=[K_{m},F_{l}]=0$ we get further

[TABLE]

This proves the assertion. ∎

Lemma 5.20** (Criterion 3).**

Let $w=us_{\alpha}$ be a Weyl group element $u=s_{m}s_{n}u^{\prime}$ and a character $\phi$ with support containing $\{\alpha_{m},\alpha_{n}\}$. Assume that the simple root $\alpha_{m}$ appears in $u(\alpha_{i})-\alpha_{m}\in\Phi$ with multiplicity zero. Then Conjecture A holds for $w$, if it holds for the shorter elements $u,s_{n}u,s_{m}u$.

Proof.

We want to explicitly construct an element in $U^{-}[s_{m}s_{n}u^{\prime}s_{i}]$ which has a character-shift with leading term $T_{u^{\prime-1}}^{-1}(F_{i})$ as asserted. , using the inductive assumption on $u=s_{m}s_{n}u^{\prime}$ as well as on $s_{n}u^{\prime}s_{i}$ and on $s_{m}u^{\prime}s_{i}$ for smaller support $\{\alpha_{n}\}$ resp. $\{\alpha_{m}\}$. For the third inductive assumption we require the stronger explicit assertion from Proposition 5.19 for $v=u^{\prime}$ that relies on the assumption of multiplicity one:

[TABLE]

Now the proof proceeds as follows: By the inductive assumption on $s_{n}u^{\prime}s_{i}$ there exists an $X\in U^{-}[s_{n}u^{\prime}s_{i}]$ such that we have $\overline{X^{\phi_{n}}}=\overline{T_{u^{\prime-1}}^{-1}(F_{i})}$ in $\mathrm{gr}U^{-}[s_{n}u^{\prime}s_{i}]_{\phi_{n}}$, or differently spoken the character-shift has leading term

[TABLE]

We may assume $X$ to be chosen in degrees $u^{\prime}(\alpha_{i})+\mathbb{Z}\alpha_{n}$, because $U^{-}[s_{n}u^{\prime}s_{i}]_{\phi_{n}}$ is still graded by $\mathbb{Z}\alpha_{n}$-cosets.

We now apply $T_{s_{m}}^{-1}$, which commutes with the $\phi_{n}$ character-shift, so we have constructed an element $T_{s_{m}}^{-1}(X)\in U^{-}[s_{m}s_{n}u^{\prime}s_{i}]$ with leading term

[TABLE]

because the only degree, which is larger then $u^{\prime}(\alpha_{i})$ and becomes smaller after reflection, is $s_{m}u^{\prime}(\alpha_{i})$, which is not in $u^{\prime}(\alpha_{i})+\mathbb{Z}\alpha_{n}$.

We now apply a $\phi_{m}$ character-shift to both sides of the formula, giving

[TABLE]

By Proposition 5.19 we know the right-hand side shift explicitly, so

[TABLE]

which concludes the proof.

∎

We remark that without multiplicity zero we cannot control on which expression we have to apply the inductive assumption in order to get precisely the root vector in the last line. For example, the character-shift could be zero or (since we apply $T_{m}^{-1}$) the initial element could even have to be chosen to be outside $U^{-}$.

We remark further, that in terms of degrees the following is hidden in the inductive assumptions of the proof, compare Example 5.15: The new PBW generator in $U^{-}[s_{n}u^{\prime}s_{i}]$ is $T^{-1}_{s_{n}}T^{-1}_{u^{\prime-1}}(F_{i})$, which is in degree $s_{n}u^{\prime}(\alpha_{i})=\beta$, while the new degree in the graded algebra is $u^{\prime}(\alpha_{i})=\beta+\alpha_{n}$. This is only possible, because (apparently, by induction) some element $Y$ in degree $\beta+\alpha_{n}=u^{\prime}(\alpha_{i})=s_{n}u^{\prime}_{1}(\alpha_{l})$ is already present in $U^{-}[s_{n}u^{\prime}]$, with character-shift $Y^{\phi_{n}}=Y+T_{s_{n}}^{-1}T^{-1}_{u^{\prime-1}}(F_{i})+\cdots$. Together with $T^{-1}_{s_{n}}T^{-1}_{u^{\prime-1}}(F_{i})$ this makes possible the linear combination $X=Y-T^{-1}_{s_{n}}T^{-1}_{u^{\prime-1}}(F_{i})$ in mixed degrees, where the leading terms in the character-shifts cancel and the next leading term is in degree $u^{\prime}(\alpha_{i})=\beta+\alpha_{n}$.

5.5. The proof of Conjecture A for type $A_{n},B_{n},C_{n},D_{n}$

The criteria above implies that Conjecture A holds for type $A_{n},B_{n},C_{n},D_{n}$, after some by-hand arguments, as we shall now discuss. Type $G_{2}$ holds trivially by rank with Criterion 1. For type $E_{6},E_{7},E_{8},F_{4}$ there are several exceptional cases, which would have to be treated by hand. Altogether a more systematic method of proof would be much preferred.

Assume first that Criterion 1 (Lemma 5.16) applies with case $A_{3}$: So in $\Phi$ we have the simple roots of the support $S=\{\alpha_{n},\alpha_{m}\}$ and a positive root $\beta$ such that $\tilde{S}=\{\alpha_{n},\beta,\alpha_{m}\}$ has inner product as in $A_{3}$, and $\beta_{i}:=u(\alpha_{i}),u^{\prime}(\alpha_{j}),\beta_{j}:=u(\alpha_{j})$ are contained in the set of positive roots generated by them. We observe $\beta\not\in\Phi(w)$, otherwise all positive roots generated by $\tilde{S}$ would be already in $\Phi(w)$. In particular $\beta_{i}\neq\beta$, so we assume without loss of generality $\beta_{i}-\alpha_{m}\in\Phi$. On the other hand we observe now that $\beta_{i}\neq\alpha_{n}+\beta+\alpha_{m}$ (the highest root in $A_{3}$), because otherwise we have $\alpha_{m}+\beta_{j}+\alpha_{n}=\beta_{i}$ or $w^{-1}(\alpha_{m})+\alpha_{j}+w^{-1}(\alpha_{n})=-\alpha_{i}$, which is absurd, because $w^{-1}(\alpha_{m}),w^{-1}(\alpha_{n})<0$ by assumption of $S\subset\Phi(w)$. It only remains $\beta_{i}=\beta+\alpha_{m}$. Overall we have reached the situation

[TABLE]

The roots $\beta$ and $\gamma$ have the property, that $\beta+\alpha_{m},\beta+\alpha_{n}\in\Phi$ and $\gamma+\alpha_{i},\beta+\alpha_{j}\in\Phi$. There is a canonical choice for such a root in any connected root system

Definition 5.21**.**

For simple roots $\alpha_{i},\alpha_{j}$ in a connected Dynkin diagram without loops, there is a unique path from $\alpha_{i}$ to $\alpha_{j}$, and summing along the path without endpoints defines a root

[TABLE]

with the property that $\rho_{ij}+\alpha_{i},\rho_{ij}+\alpha_{j}\in\Phi$. Moreover it contains $\alpha_{i},\alpha_{j}$ with multiplicity zero.

However: If $\gamma=\rho_{ij}$, then $\alpha_{n}=w(-\gamma)>0$ cannot be possible if $w(\alpha_{k})>0$ for all $\alpha_{k}\neq\alpha_{i}$. Thus, Criterion 2 (Lemma 5.18) implies Conjecture A, unless $\gamma$ contains $\alpha_{i}$ with multiplicity $\geq 1$.

Assume type $A_{n}$, then Criterion 1 (Lemma 5.16) returns only the case $A_{3}$. Since in $A_{n}$ all multiplicities are $\leq 1$, the root $\gamma$ with $\gamma+\alpha_{i}\in\Phi$ cannot contain $\alpha_{i}$ with multiplicity $\geq 1$, so Criterion 2 applies as discussed above. Alternatively we my apply Criterion 3 (Lemma 5.20), because $\beta$ with $\beta_{i}=\beta+\alpha_{m}\in\Phi$ cannot contain $\alpha_{m}$ with multiplicity $\geq 1$. Both arguments prove:

Corollary 5.22**.**

Conjecture A holds for type $A_{n}$.

Assume now type $B_{n},C_{n},D_{n}$, then Criterion 1 (Lemma 5.16) applies with the case $A_{3}$ and $B_{3}$ or $C_{3}$ or $D_{4}$, which are the possible subdiagrams of the Dynkin diagram. We first assume again case $A_{3}$ and list the possible $\beta$ explicitly:

Fact 5.23**.**

By directly inspecting the root systems $B_{n},C_{n},D_{n}$, we find that for given $\alpha_{n},\alpha_{m}$ the only positive roots $\beta$ with $\beta+\alpha_{n},\beta_{n}+\alpha_{m}\in\Phi$ are

[TABLE]

where $\alpha_{1}$ is the unique short simple root for $B_{n}$, and $\alpha_{1}$ is the unique long simple root for $C_{n}$, and $\alpha_{1},\alpha_{2}$ are the two short legs for $D_{n}$, and where the final summands depend on the type of the root system.

In the first and second case again Criterion 1 applies, because $\beta$ contains $\alpha_{m}$ with multiplicity zero. In the third case we consider $\sigma:=w(\rho_{ij})$, which is a positive root (otherwise again Criterion 2 applies, since $\rho_{ij}$ does not contain $\alpha_{i}$), and which has the property $\sigma-\beta_{i},\sigma+\beta_{j}\in\Phi$. From the explicit form of $\beta$ this only leaves $\sigma=\alpha_{m}$, so $\gamma=\rho_{ij}$. But then again Criterion 2 applies as for $A_{n}$.

Assume now type $B_{n}$ or $C_{n}$ with case $B_{3}$ or $C_{3}$. So in $\Phi$ we have the short and long simple roots of the support $S=\{\alpha_{n},\alpha_{m}\}$ and a positive (long or short) root $\beta$ such that $\tilde{S}=\{\alpha_{n},\beta,\alpha_{m}\}$ has inner product as in $B_{3}$ or $C_{3}$. We remark that by direct inspection of the root system this fixes $\beta$ to be simply the sum of all simple roots between $\alpha_{n},\alpha_{m}$, and in particular both $\alpha_{n},\alpha_{m}$ appear in $\beta$ with multiplicity zero, so Criterion 3 implies Conjecture A.

Corollary 5.24**.**

Conjecture A holds for type $B_{n},C_{n}$.

Assume now type $D_{n}$ with case $D_{4}$. So in $\Phi$ we have the simple roots of the support $S=\{\alpha_{n},\alpha_{m},\alpha_{t}\}$ and a positive root $\beta$ such that $\tilde{S}=\{\alpha_{n},\alpha_{m},\alpha_{t},\beta\}$ has inner product as in $D_{4}$. Again direct inspection of the root system shows that then $\beta$ is a sum of adjacent simple roots, and in particular contains $\alpha_{n},\alpha_{m},\alpha_{t}$ with multiplicity zero. So Criterion 3 implies Conjecture A.

Corollary 5.25**.**

Conjecture A holds for type $D_{n}$.

6. The graded algebra determining the representation theory

6.1. Conjecture B

Our ultimate hope is, that the description of the graded algebra in Conjecture A and Corollary 5.4, which we have proven in many cases determines precisely which combinations $w_{+},w_{-}$ give triangular Borel subalgebras:

Conjecture B**.**

An arbitrary triangular right coideal subalgebra $C$, i.e. of the form

[TABLE]

is a Borel subalgebra iff $w_{+}w_{-}^{\prime-1}=w_{0}$ with $\ell(w_{+})+\ell(w_{-}^{\prime-1})=\ell(w_{0})$.

Example 6.1**.**

In the homogeneous case $\phi_{+}=\phi_{-}=0$ where $w_{-}^{\prime}=w_{-}$ the condition is clear: If $\ell(w_{+})+\ell(w_{-}^{\prime-1})>\ell(w_{+}w_{-}^{\prime-1})$ then there are common roots $\alpha,-\alpha$ in $\Phi^{+}(w_{+}),\Phi^{-}(w_{-})$, producing a full quantum subgroups $U_{q}(\mathfrak{sl}_{2})$, which is surely not basic. If $\ell(w_{+}w_{-}^{\prime-1})<\ell(w_{0})$ then there are other basic algebras with larger $w_{+},w_{-}$.

Remark 6.2**.**

Already the fact that the given space is indeed an algebra puts strong (and not-yet-determined) conditions on the data. Together with maximality, we suggest that the full conjecture should first be attempted with the additional assumptions $\mathrm{supp}(\phi^{+})=\mathrm{supp}(\phi^{-})$ and $L=\mathrm{supp}(\phi^{+})^{\bot}$.

On the other hand we have so far no example of a non-triangular basic right coideal subalgebra, not contained in a triangular basic right coideal subalgebra.

The conjecture has essentially two parts, which are both representation-theoretic: Proving that $\ell(w_{-}^{\prime-1}w_{+})<\ell(w_{-}^{\prime})+\ell(w_{+})$ implies non-basic and $\ell(w_{-}^{\prime-1}w_{+})=\ell(w_{-}^{\prime})+\ell(w_{+})$ implies basic. From these two it would then follow that Conjecture B characterizes basic right coideal subalgebras that are at least maximal among all triangular basic right coideal subalgebras. We now prove the first of the first statement in general using the Conjecture A, where proven.

The second part we can so far only prove in special cases like $A_{1},A_{2}$ below or $\mathrm{supp}(\phi_{+})=\mathrm{supp}(\phi_{-})=\Phi^{+}(w_{+})\cap\Phi(w_{-})$ by explicit knowledge of the algebra. It would be desirable to find a proof using again the graded algebra and Conjecture A.

6.2. Proof of Conjecture B in one direction

We will now prove Conjecture B in one direction, by explicitly constructing higher-dimensional representations as composition factors of restricted $U_{q}(\mathfrak{g})$-representations. For example, this result allows us to prove that certain basic right coideal subalgebras we constructed [LV17] are maximal at least among the triangular basic algebras.

Lemma 6.3**.**

For any $w_{-}\in W$ let $w_{-}^{\prime}$ be as in the proven Conjecture A. Let $w_{+}\in W$, such that

[TABLE]

with $L\subset(\mathrm{supp}(\phi^{+})\cap\mathrm{supp}(\phi^{-}))^{\bot}$ is a triangular right coideal subalgebra. If $\ell(w_{-}^{\prime-1}w_{+})<\ell(w_{-}^{\prime})+\ell(w_{+})$ then $C$ is not basic.

Proof.

For an arbitrary root system $\ell(w_{-}^{\prime-1}w_{+})<\ell(w_{-}^{\prime})+\ell(w_{+})$ implies $\Phi^{+}(w_{+})\cap\Phi^{+}(w_{-}^{\prime})\neq\emptyset$, so there is a root $\mu\in\Phi^{+}(w_{+})\cap\Phi^{+}(w^{\prime-})$. By the proven Conjecture A this assumption about the graded algebra $\mathrm{gr}(C)$ means that in $C$ there are elements $E,F$ with leading terms with respect to the $\mathbb{Z}$-grading the root vectors $E_{\mu}K_{\mu}^{-1}$ and $F_{\mu}$. Note that while $E=E_{\mu}$ the element $F$ starts usually with a higher root and is not even necessarily a character-shifted root vector.

Our general strategy to construct irreducible representations of dimension $>1$ is as follows: We take some suitable finite-dimensional $U_{q}(\mathfrak{g})$ representation $V=L(\lambda)$, which we want to restrict to $C$ and decompose into irreducible composition factors, and we wish to prove not all of them are one-dimensional. The action of the commutator $[E,F]_{1}\in C$ on every one-dimensional $C$-module is trivial, so if the composition series of $C$ only contains such representations, then $[E,F]_{1}$ acts nilpotently on $V$.

But $E$ contains only terms in degree $\leq\mu$ and $F$ only terms in degree $\geq\mu$, so the commutator acts as a lower triangular matrix with diagonam entries $[E_{\mu}K_{\mu}^{-1},F_{\mu}]_{1}$. So we need to find a $U_{q}(\mathfrak{g})$ module $V=L(\lambda)$ with non-zero eigenvaules for $[E_{\mu}K_{\mu}^{-1},F_{\mu}]_{1}$, then this proves the existence of higher-dimensional irreducible composition factors.

Claim: For any root $\mu$ there exists a finite-dimensional $U_{q}(\mathfrak{g})$ representation $V$ on which the commutator $[E_{\mu},F_{\mu}]$ has a non-zero eigenvalue i.e. does not act nilpotently. For $\mu$ simple this is easily seen from the highest-weight vector in $V=L(\lambda)$ for any $(\lambda,\mu)\neq 0$ (and for $A_{n}$ in general using minuscule weights). We generalize this approach as follows:

Let $\mu=w(\alpha_{i})$ and choose any weight $\lambda$ with $(\alpha_{i},\lambda)<0$. Then in the irreducible $U_{q}(\mathfrak{g})$-module $L(\lambda)$ the weight-spaces $w(\lambda)$ and $w(\lambda-\alpha_{i})$ are both one-dimensional with basis $T_{w}(v_{\lambda}),T_{w}(F_{\alpha_{i}}v_{\lambda})$ since they are reflections of one-dimensional weight spaces. Since $E_{\mu}=T_{w}E_{\alpha_{i}}$ and $F_{\mu}=T_{w^{-1}}^{-1}F_{\alpha_{i}}$ we can evaluate the commutator $[E_{\mu},F_{\mu}]$ on $v:=T_{w}v_{\lambda}$, but things are slightly complicated by $T^{-1}_{w^{-1}}$ on $F$:

[TABLE]

where the second term vanishes by construction since $v_{\lambda}$ is a highest-weight-vector and $(T_{w}E_{\alpha_{i}})(T_{w}v_{\lambda})=T_{w}E_{\alpha_{i}}v_{\lambda}=0$. Now since the weight-spaces $w(\lambda)$ and $w(\lambda-\alpha_{i})$ are both one-dimensional and the Lusztig-automorphisms are bijective, we have scalars $a,b\neq 0$ with

[TABLE]

Since we further have

[TABLE]

it follows as claimed that $v=T_{w}v_{\lambda}$ is a non-zero eigenvector

[TABLE]

Thus as discussed above: Since $[E_{\mu},F_{\mu}]$ acts not nilpotently on the finite-dimensional $U_{q}(\mathfrak{g})$-module $L(\lambda)$, the restriction of this module to $C$ containing as leading term $E_{\mu},F_{\mu}$ cannot only have one-dimensional composition-factors. Thus we found a higher-dimensional irreducible composition factor and $C$ is not basic. ∎

7. Example $A_{2}$

We have already determined all Borel subalgebras of $U_{q}(\mathfrak{sl}_{2})$ in Section 3.2. We conclude our article by treating the next case $U_{q}(\mathfrak{sl}_{3})$ explicitly. In particular we give a completed classification of all Borel subalgebras in this case by hand, so that our conjectures and their impact can be checked against a more realistic example.

Any triangular right coideal subalgebra is of the form

[TABLE]

where the supports of the characters $\phi_{+},\phi_{-}$ consist of mutually orthogonal roots. We have proven in the previous section that $\ell(w_{-}^{\prime-1}w_{+})<\ell(w^{\prime})+\ell(w_{+})$ implies not basic and conjectured that the triangular Borel subalgebras are precisely those with $w_{-}^{\prime-1}w_{+}=w_{0}$, we expect moreover $\mathrm{supp}(\phi_{+})=\mathrm{supp}(\phi_{-})$ and by maximality $L=(\mathrm{supp}(\phi_{+})\cap\mathrm{supp}(\phi_{-}))^{\perp}$.

These expectations would lead to the following cases:

• •

$\mathrm{supp}(\phi_{+})=\mathrm{supp}(\phi_{-})=\{\}$ i.e. $\phi_{+},\phi_{i}$ trivial. These are the homogeneous Borel algebras and we have already discussed them in Section 3.3. They are reflections of $U^{-}$, explicitly

[TABLE]

• •

$\mathrm{supp}(\phi_{+})=\mathrm{supp}(\phi_{-})=\{\alpha_{1}\}$, then in particular we must have $\alpha_{1}\in\Phi^{+}[w_{\pm}]$ leaving the three cases $w_{\pm}=s_{1},s_{1}s_{2},s_{1}s_{2}s_{1}$ with $w_{\pm}^{\prime}=s_{1}w_{\pm}=1,s_{2},s_{2}s_{1}$. The relation $w_{-}^{\prime-1}w_{+}=w_{0}$ leaves the following three cases:

[TABLE]

• •

$\mathrm{supp}(\phi_{+})=\mathrm{supp}(\phi_{-})=\{\alpha_{2}\}$ leaves only cases that are isomorphic to the former ones by diagram automorphism $\alpha_{1}\leftrightarrow\alpha_{2}$.

• •

$\mathrm{supp}(\phi_{+})=\mathrm{supp}(\phi_{-})=\{\alpha_{12}\}$ leaves the three cases $w_{\pm}=s_{2}s_{1},s_{1}s_{2},s_{1}s_{2}s_{1}$ with $w_{\pm}^{\prime}=s_{12}w_{\pm}=s_{2},s_{1},1$. The relation $w_{-}^{\prime-1}w_{+}=w_{0}$ leaves up to diagram automorphism the case $w_{-}=s_{1}s_{2},w^{\prime}_{-}=s_{1},w_{+}=s_{2}s_{1}$. It is a again a reflection of the former solutions.†††The reader be advised, that reflections of right coideal subalgebras are a-priori not always right coideal subalgebras.

We shall discuss these two non-homogeneous examples of right coideal subalgebras and prove that they are basic; in the last subsection we give a by-hand classification of all basic right coideal subalgebras of $U_{q}(\mathfrak{sl}_{3})$, using similar methods as in Theorem 3.8 for $U_{q}(\mathfrak{sl}_{2})$, which shows that these are Borel subalgebras and that they are all of them.

7.1. The Borel subalgebra $U^{-}[s_{1}]_{\phi_{-}}\Bbbk[(K_{1}K_{2}^{2})^{\pm 1}]S(U^{+}[s_{1}s_{2}s_{1}])_{\phi_{+}}$

This right coideal subalgebra $C$ is generated as an algebra by

[TABLE]

where $\phi^{+}(E_{\alpha_{1}}K_{\alpha_{1}}^{-1})=\lambda$ and $\phi^{-}(F_{\alpha_{1}})=\lambda^{\prime}$ and [math] else, such that $\lambda^{+}\lambda^{-}=\frac{q^{2}}{(1-q^{2})(q-q^{-1})}$.

In the $\mathbb{Z}$-grading these elements have degree $0,0,\alpha_{1},\alpha_{2}$ and the graded algebra is in accordance with the proven Conjecture A $\mathrm{gr}(C)=\Bbbk[(K_{1}K_{2}^{2})^{\pm 1},K_{1}^{-1}]S(U^{+}[s_{2}s_{1}]$. The right coideal subalgebra $C$ contains a Weyl algebra generated by $\bar{F}_{1},\bar{E}_{1}$ from type $A_{1}\subset A_{2}$ and this injection splits by the algebra surjection sending $E_{2}K_{2}^{-1}\mapsto 0$.

We wish to prove that this algebra is basic: A direct calculation (which holds in great generality, see [LV17] Lemma 5.13) shows that for any element $X_{\mu}\in S(U^{+}[s_{1}s_{2}s_{1}])_{\phi_{+}}$ in degree $\mu$ holds

[TABLE]

Thus for any finite-dimensional representation $V$ we can consider the subspace where $E_{2}K_{2}^{-1}$ acts by zero and this is again a $C$-representation. Hence for irreducible $V$ this element $E_{2}K_{2}^{-1}$ acts by zero and the action of our algebra factorizes through the Weyl algebra in Example 3.6.

This algebra is the smallest example of a large family of basic triangular right coideal subalgebras that we constructed in [LV17] for type $A_{n}$. They are all a product of commuting copies of the quantum Weyl algebra, for every simple root in the intersection $\Phi(w_{+})\cap\Phi(w_{-})$, and then filled up as much as possible with the standard Borel subalgebra.

7.2. The Borel subalgebra $U^{-}[s_{1}s_{2}]_{\phi_{-}}\Bbbk[(K_{1}K_{2}^{2})^{\pm 1}]S(U^{+}[s_{1}s_{2}])_{\phi_{+}}$

This more complicated right coideal subalgebra $C$ is generated as an algebra by

[TABLE]

The respective graded algebra is in accordance with the proven Conjecture A

[TABLE]

Here again the characters are given by $\phi^{+}(E_{\alpha}K_{\alpha}^{-1})=\lambda$ and $\phi^{-}(F_{\alpha})=\lambda^{\prime}$ and [math] else, such that $\lambda\lambda^{\prime}=\frac{q^{2}}{(1-q^{2})(q-q^{-1})}$. The product of the appearing constants $c_{1}:=(1-q^{-2})\lambda$ and $c_{2}:=(q^{-1}-q)\lambda^{\prime}$ is

[TABLE]

We calculate the commutator relation

[TABLE]

Thus the commutators of all generators of $C$ are

[TABLE]

We prove that $C$ is basic: The algebra $C$ contains a subalgebra $\langle\bar{E}_{1},\bar{F}_{1}\rangle$ isomorphic to the Weyl algebra, which acts by Example 3.6 on every irreducible finite-dimensional representation $V_{e,f}$ via its quotient algebra $\bar{E}_{1}\bar{F}_{1}=\bar{F}_{1}\bar{E}_{1}=\frac{q^{2}}{(q-q^{-1})(1-q^{2})}$, and these are all one-dimensional. But the relation reduces the commutator relation 5 to

[TABLE]

Consider now for any irreducible finite-dimensional $C$-representation $V$. Since $K$ is central it acts by a scalar. Let $V_{e,f}$ be a irreducible Weyl algebra representation, which is one-dimensional. By the $q$-commutators 2 and 3 the generators $\bar{E}_{12},\bar{F}_{12}$ map $V_{e,f}$ to some subrepresentation $V_{qe,q^{-1}f}$ resp. $V_{q^{-1}e,qf}$. Since $ef\neq 0$ and $q$ is not a root of unity, finite dimension proves that $\bar{E}_{12},\bar{F}_{12}$ act nilpotently. Let $V_{e^{\prime},f^{\prime}}$ be such that $\bar{E}_{12}$ acts by zero, then by the additional relation 6 shows that $\bar{F}_{12}$ preserves $V_{e^{\prime},f^{\prime}}$ as well (and it acts also by zero). Hence any irreducible finite-dimensional $C$-representation is one-dimensional.

7.3. Classification Result

From our proven direction of Conjecture B it follows that the basic right coideal subalgebras introduced above are maximal among all triangular basic right coideal subalgebras. With similar techniques as for $U_{q}(\mathfrak{sl}_{2})$ in Theorem 3.8 one can show:

Theorem 7.1** ([Vocke16] Chapter 10).**

The triangular basic right coideal subalgebras above

[TABLE]

are Borel subalgebras and these are all Borel subalgebras of $U_{q}(\mathfrak{sl}_{3})$ up to reflection and diagram automorphism.

The proof idea for both assertions is that from [Vocke16] Theorem 4.11 and Lemma 3.5 we have for any right coideal subalgebra in $U_{q}(\mathfrak{g})$ a generating system of elements, which have a unique leading term that is a root vector. Then one has to study combinations of arbitrary elements and test them on (minuscule) representations of $U_{q}(\mathfrak{sl}_{3})$ to rule basic out, or to show that the combination is contained in a triangular right coideal subalgebra, because we know our examples are the maximal ones among them. These results beyond the results on Conjecture A and B are very ad-hoc.

7.4. Induction of one-dimensional characters

We want to finally calculate the induced representations in the two nontrivial Borel algebras above in $U_{q}(\mathfrak{sl}_{3})$ as in Section 4.

Let $C=U^{-}[s_{1}]_{\phi_{-}}\Bbbk[(K_{1}K_{2}^{2})^{\pm 1}]S(U^{+}[s_{1}s_{2}s_{1}])_{\phi_{+}}$. We have seen that each irreducible representation is of the form $V_{e_{1},f_{1},k_{122}}$, which is one-dimensional with action given by scalars $e_{1},f_{1},k_{122}\in\Bbbk^{\times}$ with $e_{1}f_{1}=\frac{q^{2}}{(q-q^{-1})(1-q^{2})}$

[TABLE]

Since $C$ has a PBW-basis, we can calculate the induced module to be

[TABLE]

This is (up to $K_{2}^{\epsilon}$, which depends on the chosen lattices) the right coideal subalgebra $\tilde{C}=U^{-}[s_{2}s_{1}]\Bbbk[K_{1}]$, acting on itself by left-multiplication, and this action is extended to $U_{q}(\mathfrak{sl}_{3})$ acting on $\tilde{C}$. A similar result follows easily whenever $\Phi^{+}(w_{-})\cap\Phi^{+}(w_{+})=\mathrm{supp}$ as we show in [LV17].

Let $C=U^{-}[s_{1}s_{2}]_{\phi_{-}}\Bbbk[(K_{1}K_{2}^{2})^{\pm 1}]S(U^{+}[s_{1}s_{2}])_{\phi_{+}}$. We have seen that each irreducible representation is of the form $V_{e_{1},f_{1},k_{122}}$, which is one-dimensional with action given by scalars $e_{1},f_{1},k_{122}\in\Bbbk^{\times}$ with $e_{1}f_{1}=\frac{q^{2}}{(q-q^{-1})(1-q^{2})}$

[TABLE]

Since $C$ has a PBW-basis, we can calculate the induced module to be

[TABLE]

This is (up to $K_{2}^{\epsilon}$, which depends on the chosen lattices) the Hopf subalgebra $\tilde{C}=U_{q}(\mathfrak{sl}_{2})$, acting on itself by left-multiplication, and this action is extended to $U_{q}(\mathfrak{sl}_{3})$ acting on $\tilde{C}$.

In both cases our results in Section 4 state that any finite-dimensional $U_{q}(\mathfrak{sl}_{3})$ representation appears as quotient of the induced representation for some specific values $(e_{1},f_{1},k_{122})$. The tool in Lemma 4.9 shows again that for generic values the induced representations are irreducible.

8. Outlook

Our efforts are far from being concluded. We wish to point out from our perspective difficult points that need to be resolved in future research:

1. (1)

Conjecture A should be proven for all $\mathfrak{g}$. That is, $\alpha_{i}=\alpha_{j}$ in the induction step for any $s_{i}w$. As we have proven in Sec. 5.5, this holds by a combination of three criteria from root system combinatorics to explicit calculations in most cases by uniques, except a few exceptions in $E_{6},E_{7},E_{8}$ with high multiplicity. These could in principal be treated by hand, by computing the character-shifts in non-multiplicity-zero cases of Criterion 3. However, it would be much more satisfying to have a systematic proof. 2. (2)

Conjecture B should be proven in the open direction, and the graded algebra $\mathrm{gr}(C)$ together with Conjecture A should be used more systematically. 3. (3)

Similarly, we would expect that the graded algebra $\mathrm{gr}(C)$ gives much information about the structure of the induced module. In essence, one would want to use the preimage of $\mathrm{gr}(C)^{0}$ much like the Cartan part for a usual Verma module. On should also try to prove standard facts, for example that the tops are irreducible modules. 4. (4)

The explicit family of triangular right coideal subalgebras in $U_{q}(\mathfrak{sl}_{n+1})$ (conjecturally Borel) in [LV17] consisting of a positive part and several quantum Weyl algebras. It seems to be a rather uniform construction, also in higher depth and for arbitrary $\mathfrak{g}$, which should be studied. All other examples we know (e.g. the last example in $\mathfrak{sl}_{3}$) contain extensions of Weyl algebras by Weyl algebras, and maybe they can be inductively treated. Are there completely different cases? 5. (5)

Beyond triangular Borel algebras, we have observed in small examples (but do not dare to conjecture) that for a possibly non-triangular basic right coideal subalgebras, there always seems to be a larger triangular right coideal subalgebras, which is still basic. This would imply all Borel algebras are triangular. If not, than a counter-example would be extremely interesting. Up to now, the only access we have is to use the generator theorem in [Vocke16] for arbitrary coideal subalgebras and then combine possible generators by hand.

Moreover, the following question are from our perspective interesting with respect to applications:

1. (1)

We would find it very interesting to classify Borel subalgebras for small quantum groups when $q$ is a root of unity. 2. (2)

For any Borel subalgebra $B$, do the induced modules produce a good analog of the tensor category $\mathcal{O}_{B}$? 3. (3)

The fact that we use coideal subalgebra means that the category of representations of $C$ is a module category over $\mathrm{Rep}(U_{q}(\mathfrak{g}))$. What additional structure is the implication of this fact? One might expect some weak link between the category $\mathcal{O}_{B}$ and usual category $\mathcal{O}$, maybe an invertible bimodule category as in [LP17]? 4. (4)

The induced modules appear for $\mathfrak{sl}_{2}$ in [Schm96, Tesch01] in the context of non-compact quantum group. For $\mathfrak{g}$, are there corresponding families of modules with these additional analytic data? (for example, for $\mathfrak{sl}_{2}$ replacing $\mathbb{C}[K,K^{-1}]$ by functions on the torus).

On the other hand, is our construction related to the non-standard Borel subalgebras of affine Lie algebras [Fut94, Cox94] via Kazhdan-Lusztig correspondence?