ZZMP-HH 17-17Hamburger Beitr¨age zur Mathematik Nr. 662May 2019
On Borel subalgebras of quantum groups
Simon Lentner ∗ , University of HamburgKarolina Vocke, University of Oxford Abstract.
For a quantum group, we study those right coideal subalgebras,for which all irreducible representations are one-dimensional. If a right coidealsubalgebra 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 newunfamiliar Borel subalgebras, for example ones containing copies of the quantumWeyl algebra. Given a Borel subalgebra, we study its induced (Verma-)modulesand prove among others that they have all irreducible finite-dimensional mod-ules as quotients. We then give structural results using the graded algebra, whichin particular leads to a conjectural formula for all triangular Borel subalge-bras, which we partly prove. As examples, we determine all Borel subalgebrasof U q ( sl ) and U q ( sl ) and discuss the induced modules. Contents
1. Introduction 22. Preliminaries 62.1. Quantum groups 62.2. Quantum group representation theory 83. Borel subalgebras 83.1. Main Definitions 83.2. Example A A A n , B n , C n , D n A U − [ s ] φ − k [( K K ) ± ] S ( U + [ s s s ]) φ + U − [ s s ] φ − k [( K K ) ± ] S ( U + [ s s ]) φ + ∗ Corresponding author: [email protected] a r X i v : . [ m a t h . QA ] M a y Introduction
Borel subalgebras are an essential element in the structure theory of a semisim-ple Lie algebra g , and the representations induced from a Borel subalgebra are anessential element in the representation theory of g . By Lie’s theorem, the Borelsubalgebras in g are precisely the maximally solvable subalgebras, and all Borelsubalgebras are conjugate, in particular isomorphic.I. Heckenberger has asked the analogous question for a quantum group U q ( g ),namely to construct and classify those right coideal subalgebras C of U q ( g ) thatare 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 thatfor quantum groups there are additional families of such Borel subalgebras, whichare not present g . An example already for sl and q generic is given below.The main motivation for studying Borel subalgebras of quantum groups is themore general goal to understand the set of all coideal subalgebras, which is con-sidered a main problem in the area of quantum groups, with considerable progressmade in [Let99, KS08, HS09, HK11a, HK11b]. Main examples are the quantumsymmetric 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 Liesubalgebra of g decomposes into a solvable Lie algebra and a semisimple Lie alge-bra. 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 sub-algebras 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 allowsus to repeat the standard construction of Verma modules, namely inducing up aone-dimensional representation of C to a representation of U q ( g ). Our new Borelsubalgebras lead to new families of quantum group representations, in which typ-ically the Cartan part acts non-diagonalizably, but which otherwise share manyfeatures of usual Verma modules, for example having the finite-dimensional irre-ducible modules as quotients. For sl , these new modules have already appearedas 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 conceivablethat our construction is related to this construction via Kazhdan-Lusztig corre-spondence.The goal of this article is to initiate the study of these Borel subalgebras. Forone, by constructing examples and give complete classifications for g = sl , sl . Conceptually, we introduce structural tools and conjecture a general formula forBorel subalgebras, which may or may not be exhaustive. We can prove parts of thisconjecture and check it in further examples. Moreover we prove general propertiesof the induced modules and construct then in example classes.As a main structural tool for both questions, we propose to study the gradedalgebra gr( C ) of a general coideal subalgebra C . For a coideal subalgebra C in U + q ( g ), which where classified in [HK11b] as character-shifts, we give for gr( C )a conjectural formula (Conjecture A), which may be of independent interest andwhich also implies a formula for gr( C ) for triangular C in U q ( 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 representationtheory of C . From these considerations we derive an explicit conjectural formula(Conjecture B) for all Borel subalgebras with a triangular decomposition. We areable to prove one direction of this assertion i.e. we obtain a necessary form for allbasic triangular coideal subalgebras. For example, this shows that basic coidealsubalgebras of U q ( sl n +1 ) we constructed in [LV17] are maximal among all triangu-lar 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 ( g ), where g is a finite-dimensional semisimple Lie algebra and q ∈ k × is not a root of unity, and our basefield k is an algebraically closed field of characteristic 0.In Section 2, we collect preliminaries on quantum groups and their representa-tion theory. We also briefly review the current state of the classification of rightcoideal subalgebras of quantum groups: In [HS09], the homogeneous graded rightcoideal subalgebras of U ≤ q ( g ) are shown to be in bijection with Weyl group el-ements w ∈ W via U − q [ w ] U . In [HK11a], the homogeneous graded right coidealsubalgebras of U q ( g ) are classified as suitable combinations U − q [ w − ] U S ( U + q [ w + ]).In [HK11b], arbitrary right coideal subalgebras C ⊂ U ≤ q ( g ), with C a Hopf al-gebra, are classified to be so-called character-shifts U − q [ w ] φ k [ L ] with a suitablesubalgebra of the Cartan part k [ L ] ⊂ U . The classification of arbitrary rightcoideal subalgebras in U q ( g ) is still a difficult open problem. We hope that by re-stricting our attention to basic right coideal subalgebras, we make a further step.In Section 3, we give our definition of Borel subalgebras and discuss thoroughlytwo examples: First we classify all Borel subalgebras of U q ( sl ). We find the twostandard Borel subalgebras U ≥ and U ≤ , which are already present in the Liealgebra sl , and we find a new family of non-homogeneous Borel subalgebras B λ,λ (cid:48) : Example (3.6) . In U q ( sl ) there is a family of non-homogeneous Borel subalgebras B λ,λ (cid:48) with algebra generators ¯ E := EK − + λK − , ¯ F := F + λ (cid:48) K − , for λλ (cid:48) = q (1 − q )( q − q − ) . Because of the denominator of λλ (cid:48) , the subalgebras B λ,λ (cid:48) are not contained in theintegral form of U q ( sl ) and do not specialize to q = 1 . They interpolate betweenthe two standard Borel subalgebras U ≥ and U ≤ for λ → and λ (cid:48) → .These Borel subalgebras are isomorphic to one another for different values of λ and λ (cid:48) by a Hopf automorphism of U q ( g ) and they are isomorphic as an algebra tothe quantum Weyl algebra (cid:104) x, y (cid:105) / ( xy − q yx − . As a second initial class of examples, we show in Section 3.3 that, as expected,the Borel subalgebras of U q ( g ) that are homogeneous with respect to the N n -grading are precisely the standard Borel subalgebras.In Section 4, we turn our attention to representation theory: One importantapplication of a Borel subalgebra b of a Lie algebra g is the construction of Vermamodules V ( λ ), which are induced modules g ⊗ b χ of a one-dimensional characteron a Borel subalgebra χ : b → k , which is given by a weight λ ∈ h ∗ .Analogously, we construct an infinite-dimensional U q ( g )-modules V ( B, χ ) forany Borel subalgebra B in our sense and any one-dimensional character χ . We thenprove some general properties, in particular that for any given Borel subalgebra B all finite-dimensional irreducible representations L ( λ ) of U q ( g ) appear as quotientsof induced modules of one-dimensional character χ of B .Then we study exemplary the induced modules for the unfamiliar Borel sub-algebras B λ,λ (cid:48) in U q ( sl ). As an algebra this is a quantum Weyl algebra, andone-dimensional representations are parametrized by complex numbers e, f with ef = λλ (cid:48) . The resulting infinite-dimensional U q ( sl )-modules are isomorphic to theCartan part U = k [ K ± ], and U acts non-diagonalizably by left-multiplication.We determine all sub- and quotient modules. We find that for a generic character χ the induced module is irreducible, while for a discrete set of characters χ thereis a unique non-trivial quotient, which is isomorphic to L ( λ ).In Section 5, we restrict our attention to triangular right coideal subalgebras C = C ≤ C ≥ . By the classification in [HK11b], both parts are character-shifts, so C = U − q [ w − ] φ − k [ L ] S ( U + q [ w ]) φ + with unknown compatibility conditions for the data ( w ± , φ ± , L ). On the otherhand gr( C ) is (up to localization, see below) a coideal subalgebra classified in[HS09][HK11a] in terms of Weyl group elements w (cid:48)± . The main question is to give a formula for w (cid:48)± depending on the data ( w ± , φ ± , L ). Motivated by the degree distri-bution and example calculations, we conjecture a general formula as Conjecture A.The remainder of this section is devoted to derive different criteria that implyConjecture 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 sys-tematically, we find that our criteria imply Conjecture A for all coideal subalgebrasin these cases, and quite trivially also for G . For E , E , E , F there are severalexceptional cases, which are not covered by our criteria and which at our currentstate could only be settled by tedious computer calculations or (preferably) newstructural ideas. We now sketch and discuss the structure of our proof: • In Section 5.1, we formulate Conjecture A and list the consequences forarbitrary coideal subalgebras. We illustrate it by treating a first examplein type A . A technical complication is that in general gr( C ) is not a Hopfalgebra (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 thelocalization of gr( C ) on gr( C ) , which captures the essential information. • In Section 5.2, the localization is discussed. The main output of this con-sideration is Corollary 5.13, which compares the growth of the localizationsand results in the formula (cid:96) ( w ) = rank( ˜ G ) + (cid:96) ( w (cid:48) ). • In Section 5.3, an inductive proof for Conjecture A is set up. For Weyl groupelements w = us α i determining C , the Weyl group element determining thelocalization of gr( C ) is prolonged to w (cid:48) = u (cid:48) s α j or w (cid:48) = u (cid:48) depending onthe case. It remains to show that α i = α j . • In Section 5.4, we give three criteria that prove α i = α j in the inductionstep. Criterion 1 uses the relation between α i and α j to determine all criticalsituations up to reflection. Criterion 2 proves the induction step provided w does not have a unique ending. Criterion 3 proves the induction step bydirect calculation of the character shift of a reflection T s m , provided α m appears in the root with multiplicity zero. This calculation uses a slightgeneralization of [Jan96] 8.14 (6) from the case of a simple root to a rootwith 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 λ,λ (cid:48) above, the graded algebra is as predicted by Conjecture Awith w (cid:48) = 1 for w = s :gr( B λ,λ (cid:48) ) = (cid:104) K − (cid:105)(cid:104) S ( E ) (cid:105) = U − [1] C [ K − ] S ( U − [ s ])In Section 6 we start to apply Conjecture A to control representation-theoreticproperties of a triangular right coideal subalgebra C . We formulate as ConjectureB a complete description of the data ( w ± , φ ± , L ) that lead to triangular Borelsubalgebras of U q ( g ), namely precisely those for which gr( C ) localizes to a Borelsubalgebra. 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 notbasic is to find higher-dimensional C -representations as composition factors in therestriction of a suitable U q ( g )-representation L ( λ ).In Section 7 we thoroughly treat the case U q ( sl ), and we give a complete clas-sification of all Borel subalgebras. We find three types of Borel subalgebras, whichcan all be given in terms of generators and relations: • The standard Borel subalgebras. • Borel subalgebras that come from the Borel subalgebras B λ,λ (cid:48) in U q ( sl ),together with a remaining standard Borel subalgebra. This is the smallestexample of a family we constructed [LV17]. • Another family of Borel subalgebras that consist of an extension of a Weylalgebra by another Weyl algebra.In all these cases we also determine the induced representations.
Acknowledgement.
We are very thankful to Istvan Heckenberger for giving veryvaluable impulses at several stages of this paper. We are also thankful for his hospi-tality in Marburg, which was made possible by the Humboldt Institut Partnerschaft.We thank Yorck Sommerhuser for comments and literature hints in the final stageof this article. The first author also receives partial support from the RTG 1670. Preliminaries
Quantum groups.
Throughout this article g is a finite-dimensional semisim-ple Lie algebra of rank n . Associated to this datum is a root system.We denote a fixed set of positive simple roots by Π = { α , . . . α n } and the cor-responding set of all positive roots by Φ + . The simple roots are a basis of the rootlattice Λ with bilinear form ( , ) and Cartan matrix c ij = 2 ( α i ,α j )( α i ,α i ) . Let W be thefinite Weyl group acting on the lattice Λ by reflections, generated by the simplereflections s i on the hyperplanes α ⊥ i .Let k be an algebraically closed field of characteristic 0, and let q ∈ k × be nota root of unity. We consider the quantum group U = U q ( g ), and for the followingstandard facts we refer the reader e.g. to [Jan96]. The algebra U is a deformationof the universal enveloping algebra of the Lie algebra U ( g ). It is generated byelements E α , F α and K ± α with α ∈ Π and graded by Λ.There is a triangular decomposition U = U + U U − into subalgebras U + , U , U − generated by the E, K, F , respectively, and we have Hopf subalgebras U ≥ = U U + and U ≤ = U U − corresponding to Borel subalgebras of g . The Cartan part U = C [ K ± α , . . . , K ± α n ] is the group ring of the root lattice Λ ∼ = Z n .The algebra U can be endowed with the structure of a Hopf algebra, and thisendows the category of representations Rep( U ) of the quantum group with a tensorproduct ⊗ k . We write [ X, Y ] λ = XY − λY X for any λ ∈ k and X, Y ∈ U q ( g ). For every Weyl group element w ∈ W there is an algebra automorphisms T w on U q ( g ) due to Lusztig. For any fixed choice of a reduced expression of the longestelement of the Weyl group w = s α i · · · s α i(cid:96) ( w with (cid:96) ( w ) = | Φ + | we get anenumeration of the set of positive roots β k = s α i · · · s α ik − α i k . Using Lusztig’sautomorphisms this defines root vectors E β k = T s αi ··· s αik − ( E α ik ) and a PBW basisfor 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 coidealsubalgebra if ∆( C ) ⊂ C ⊗ H .We call a right coideal subalgebra C ⊂ U q ( g ) homogeneous if U ⊂ C . In par-ticular C is then homogeneous with respect to the Λ -grading. Example 2.2.
Any Hopf subalgebra is in particular a right coideal subalgebra.For example U + U resp. U − U generated by the E α resp. F α and K α , K − α arehomogeneous right coideal subalgebras. The subalgebras U − and S ( U + ) are rightcoideal 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 ∈ W there is a right coidealsubalgebra U + [ w ] U , where U + [ w ] is generated by the root vectors E β i for all β i inthe following subset of roots Φ + ( w ) = { α ∈ Φ + | w − α < } = { β i | i ∈ { , . . . , (cid:96) ( w ) }} In particular | Φ + ( w ) | = (cid:96) ( w ) , and v < w ⇔ Φ + ( v ) ⊂ Φ + ( w ) The space U + [ w ] does not depend on the choice of a reduced expression of w .Conversely, every homogeneous right coideal subalgebra C ⊂ U + q ( g ) U is of thisform for some w . Theorem 2.4 ([HK11a] Theorem 3.8) . Every homogeneous right coideal subalgebra C ⊂ U q ( g ) is of the form C = U + [ w ] U U − [ v ] for a certain subset of pairs v, w ∈ W . Non-homogeneous right coideal subalgebras are only classified in U ± U : Theorem 2.5 ([HK11b] Theorem 2.15) . For w ∈ W , let φ : U − [ w ] → k be aone-dimensional character and define supp( φ ) := { β ∈ Λ | ∃ x β ∈ U − q [ w ] with φ ( x β ) (cid:54) = 0 } , which consists of mutually orthogonal simple roots. Take any subgroup L ⊂ supp( φ ) ⊥ ,then there exists a character-shifted right coideal subalgebra U − [ w ] φ := { φ ( x (1) ) x (2) | ∀ x ∈ U − [ w ] } and a right coideal subalgebra U − [ w ] φ k [ L ] with group ring k [ L ] = k [ L ] ⊂ U .Conversely, every right coideal subalgebra C ⊂ U − U with C a Hopf algebrasis of this form for some datum ( w, φ, L ) . Quantum group representation theory.
A main idea for Lie algebra andquantum group representations is to take a large commutative subalgebra h ⊂ g (Cartan part) and simultaneously diagonalize the action. The possible eigenvalues h → k are called weights. Definition 2.6 (Verma module) . Every one-dimensional character χ of U = k [Λ] can be extended trivially to U + U . Consider the induced U -representation V ( χ ) = U ⊗ U + U k χ ∼ = U − v λ generated by a highest weight vector v λ := 1 ⊗ χ with E α v λ = 0 and K α v λ = χ ( K α ) v λ for all α ∈ Π . The module V ( χ ) has a unique irreducible quotient module L ( χ ) .In particular for a weight λ of g , the Verma module V ( λ ) of type +1 is theinduced representation associated to the character χ ( K α ) = q ( λ,α ) . If λ is an in-tegral dominant weight then L ( λ ) is finite-dimensional. Every finite-dimensionalirreducible module is the quotient of the induced module for a unique character χ . Recall that we always assume q is not a root of unity. In this case, the categoryof finite-dimensional U q ( g )-modules is semisimple, and resembles the case U ( g ). Example 2.7.
For every n ≥ there are two irreducible U q ( sl ) -modules L ( n, ± ) of dimension n + 1 with basis m , m , . . . , m n such that Km i = ± q n − i m i F m i = (cid:40) m i +1 , for i < n, , for i = nEm i = (cid:40) ± [ i ] q [ n + 1 − i ] q m i − , for i > , , for i = 0 Every finite-dimensional irreducible U q ( sl ) -module is of this form. Borel subalgebras
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 irreduciblerepresentations are one-dimensional.
Example 3.2.
Finite dimensional algebras are basic if they are basic in the usualsense i.e. A/ Rad( A ) ∼ = k 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 ( g ) a Borel subalgebra if it is basic and it is maximal (with respect to inclusion) amongall basic right coideal subalgebras.
The idea of this notion is to generalizes the characterization of a Borel subalge-bra as the maximal solvable Lie subalgebra. A big simplification is the followingadditional technical assumption:
Definition 3.4.
We call a right coideal subalgebra C of U q ( g ) triangular , if C = C ≤ C ≥ with C ≤ = C ∩ U ≤ and C ≥ = C ∩ U ≥ . There are many examples of non-triangular right coideal subalgebras, but wehave no example of a non-triangular Borel subalgebra.
Example 3.5.
All homogeneous Borel subalgebras are isomorphic to the standardBorel subalgebra U + U . We will prove this in Section 3.3. Example 3.6. In U q ( sl ) there is a family of non-homogeneous Borel subalgebras B λ,λ (cid:48) with algebra generators ¯ E := EK − + λK − , ¯ F := F + λ (cid:48) K − , λλ (cid:48) = q (1 − q )( q − q − ) We will see below that B λ,λ is basic, because it is isomorphic to the quantum Weylalgebra. The maximality will be proven in the next section. The condition on λ, λ (cid:48) becomes clear if we calculate[ ¯ E, ¯ F ] q = [ EK − + λK − , F + λ (cid:48) K − ] q = K − K − q − q − q K − + λλ (cid:48) K − K − (1 − q )= q q − q − · λ, λ (cid:48) we have K − in the subalgebra B λ,λ (cid:48) , and hence es-sentially all of U q ( g ), which is surely not basic. On the other hand it is known: Lemma 3.7.
The quantum Weyl algebra (cid:104)
X, Y (cid:105) / ( XY − q Y X − c is basic.Proof. Let V be a finite dimensional irreducible representation. Consider the eigen-vector v of the element T := Y X with eigenvalue t (which is related to the quantumCasimir). One can easily see, that Y v is an eigenvector with the eigenvalue qt + 1: Y X ( Y v ) = Y ( qY X + 1) v = ( qt + 1) Y v
Similarly, one can show that Xv is an eigenvector of T with eigenvalue q ( t − Y X ( Xv ) = 1 q ( XY X − X ) v = 1 q ( t − Xv Thus the eigenvectors of T are a basis of V , as V is irreducible. On the other handfor each i there are eigenvectors Y i v of T . As V is finite dimensional, they cannothave 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 = − q .With this t we get: XY v = ( qY X + 1) v = (cid:18) − q q + 1 (cid:19) v = ( q + 1 − q )1 − q v = tv 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 finitedimensional irreducible representation is one-dimensional, as asserted. (cid:3) Example A . The root system A of rank 1 has Φ = { α, − α } with ( α, α ) = 2.The corresponding quantum group U q ( sl ) is generated by E, F, K, K − such that[ E α , F α ] = K α − K − α q − q − [ E α , K α ] q − = [ F α , K α ] q = 0 Theorem 3.8.
The Borel subalgebras of U q ( sl ) are • The standard Borel algebras U ≥ and U ≤ . • For any pair of scalars with λλ (cid:48) = q (1 − q )( q − q − ) the algebra in Example 3.6 B λ,λ (cid:48) := (cid:104) EK − + λK − , F + λ (cid:48) K − (cid:105) Proof.
We know from the previous lemma that all algebras in the assertion arebasic 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 coidealsubalgebra in U q ( sl ) has a set of generators of the form K i , EK − + λ E K − , F + λ F K − , EK − + c F F + c K K − for some constants c F , c K , λ F , λ E (cid:54) = 0. We now checkfor all combinations of two such elements which algebra they roughly generate: K i EK − + λ (cid:48) E K − F + λ (cid:48) F K − EK − + c (cid:48) F F + c (cid:48) K K − K i U U ≥ U ≤ UEK − + λ E K − U ≥ U ≥ ( λ E (cid:54) = λ (cid:48) E ) B λ,λ (cid:48) B λ,λ (cid:48) F + λ F K − U ≤ B λ,λ (cid:48) U ≤ ( λ E (cid:54) = λ (cid:48) E ) B λ,λ (cid:48) EK − + c F F + c K K − U B λ,λ (cid:48) B λ,λ (cid:48) B λ,λ (cid:48) ( c F : c K (cid:54) = c (cid:48) F : c (cid:48) K )Most entries in the table follow simply by subtracting suitable multiples of oneanother (and ignoring possible K -powers). The entry involving EK − + c F F + c K K − and K i comes from the fact that K commutes differently with E, F , sosuitable commutators return the individual summands, again up to K -powers. Clearly U is not basic. When the elements generate some B λ,λ (cid:48) but fail thecondition on λλ (cid:48) , then as we saw their commutator [ EK − + λK − , F + λ (cid:48) K − ] q is a linear combination of 1 and K − with non-zero coefficients, so the generatedalgebra contains K − . This situation is very similar to U , and indeed since everyirreducible U -module L ( k, ± ) restricts to an irreducible over this algebra, thus weobtain irreducible higher-dimensional representations.So the only remaining cases are when these two generators are equal, or whenthey generate U ≥ , U ≤ or when they generate B λ,λ (cid:48) . (cid:3) Example 3.9.
In upcoming proofs we will frequently use restrictions of suitable U q ( g ) -modules to construct higher-dimensional representations of some B in ques-tion. The restrictions are in general neither irreducible nor semisimple. For a Borelsubalgebra all composition factors need to be one-dimensional.As an explicit example, let L ( λ ) , λ = α/ be the 2-dimensional module for U q ( sl ) with basis x := v λ , x := v λ − α . Then the elements of the Borel subalgebra B λ,λ (cid:48) act as follows: ( EK − + λ E K − ) .x = λq − x ( EK − + λ E K − ) .x = λqx + qx ( F + λ (cid:48) K − ) .x = λ (cid:48) q − x + x ( F + λ (cid:48) K − ) .x = λ (cid:48) qx Hence L ( λ ) restricted to B λ,λ (cid:48) has a one-dimensional submodule (cid:104) x + λ (1 − q − ) x (cid:105) ,where the elements EK − + λK − , F + λ (cid:48) K − act with eigenvalues λq, λ (cid:48) q − , anda one-dimensional quotient L ( λ ) / ( x + λ (1 − q − ) x ) with eigenvalues λq − , λ (cid:48) q . Example: Homogeneous Borel subalgebras.
We want to prove that allhomogeneous Borel subalgebras of U q ( g ) are standard Borel subalgebras i.e. reflec-tions via T w of U ≤ . We start by proving that they are basic: Lemma 3.10.
For any Weyl group element w ∈ W consider the right coidealsubalgebra U − [ w ] and the restriction of the counit (cid:15) : U → k . Then any finite-dimensional irreducible representation V on which all elements in U − [ w ] ∩ ker( (cid:15) ) act nilpotently is one-dimensional and hence the trivial representation k (cid:15) .Proof. From Chapter 6 in [HS09] we know, that U − [ w ] is isomorphic to a smashproduct k [ F α ] T α ( U − [ s α w ]) with s α < w . We prove our claim by induction on thelength (cid:96) ( w ):For w = 1 the claim is certainly true. Let the assertion be proven for v ∈ W andconsider the case w := s α v with (cid:96) ( w ) = (cid:96) ( v ) + 1. Assume a representation V onwhich U − [ w ] ∩ ker( (cid:15) ) acts nilpotently. Restricting to the subalgebra T α ( U − [ v ]) ⊂ U − [ w ] induction provides that all composition factors are one-dimensional k (cid:15) . Inparticular there is a common trivial eigenvector of this subalgebra h.v = (cid:15) ( h ) v .Let V ⊂ V be the nontrivial subspace V = { v ∈ V | h.v = (cid:15) ( h ) v ∀ h ∈ T α ( U − [ v ]) } This is clearly a T α ( U − [ v ])-subrepresentation. Moreover we know that ad F α actson T α ( U − [ v ]) in the smash product, so for any X µ ∈ T α ( U − [ v ]) µ X µ F α .v = q ( µ,α ) F α X µ .v + Y.v with some Y ∈ T α ( U − [ v ])It follows that F α preserves V and because V was assumed irreducible we get V = V . As F α acts on V nilpotently it has an eigenvector v (cid:48) with eigenvalue 0,which is then a one-dimensional trivial representation of U − [ w ]. (cid:3) In particular we know that all elements F α ∈ U q ( g ) act nilpotently on V (see[Jan96] Proposition 5.1). So the previous lemma shows that all irreducible subquo-tients of U q ( g )-modules are trivial one-dimensional. For an arbitrary representationof U − [ w ] this is certainly not true, as already the example U − [ s α ] = k [ F α ] 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 ∈ W , L a subgroup of Λ and thecorresponding right coideal subalgebra C = k [ L ] U − [ w ] with the property, that forall µ ∈ Φ + ( w ) there exists a ν ∈ L such that ( µ, ν ) (cid:54) = 0 , then C is basic.In particular U U − [ w ] is basic, as well as the right coideal subalgebra U U + [ w ] .Proof. The proof is similar to the proof of the Lemma 3.10: We show, that eachfinite dimensional irreducible representation is one-dimensional and that U − [ w ]acts on all one-dimensional representations by (cid:15) .As k [ L ] is abelian, the claim holds for w = 1. Again we consider inductively therestriction of a k [ L ] U − [ w ]-module to the subalgebra T s α ( U − [ v ]) where by inductionwe find a common eigenvector v with E β .v = 0 and K β .v = χ ( K β ) v for somecharacter χ : L → k × extended trivially to k [ L ] T s α ( U − [ v ]). Again we consider thenontrivial subspace V = { v ∈ V | h.v = χ ( h ) v ∀ h ∈ T s α ( U − [ v ]) } and again the ad F α -stability of T s α ( U − [ v ]) in the smash product, and the assump-tion that T s α ( U − [ v ]) acts trivially implies that V is a U − [ w ]-submodule.Let v (cid:48) be an eigenvector of F α , then the new issue in this proof is whether theeigenvalue λ may be nontrivial. But using the assumed element K ν with ( α, ν ) (cid:54) = 0with commutator relation K ν F α = q − ( α,ν ) F α K ν with q not a root of unity impliesthen an infinite family of eigenvectors with distinct eigenvalues q − n ( α,ν ) , which is acontradiction. Hence F α acts on V on the eigenvector by 0 and we have again founda one-dimensional subrepresentation with trivial action of U − [ w ] as asserted. (cid:3) Theorem 3.12.
The basic right coideal subalgebra U U − [ w ] is maximal with thisproperty and hence a Borel subalgebra. We call it the standard Borel subalgebra .Conversely each homogeneous Borel subalgebra B is as algebra isomorphic to thestandard Borel subalgebra via some T w . Explicitly B = U + [ w ] U U − [ w − w ] . Proof.
Since every right coideal subalgebra B ⊃ U − [ w ] U is by definition homoge-neous it is sufficient for both assertions to prove that any homogeneous basic rightcoideal subalgebra is contained in some B = T w ( U − U ).By [HK11a] Prop. 2.1 every homogeneous right coideal subalgebra B is triangu-lar and together with the classification of homogeneous right coideal subalgebrasof U ± q in [HS09] Thm 7.3 they are of the form: B = U − [ v ] U S ( U + [ w ])We are finished if we can prove from B being basic that Φ + ( w ) ∩ Φ + ( v ) = {} .So assume to the contrary that there is µ ∈ Φ + ( w ) ∩ Φ + ( v ), so the root vectors E µ K − µ , F µ ∈ B and hence K − µ ∈ B . Let λ be a dominant integral weight with( λ, µ ) (cid:54) = 0, then there exists in the U q ( g )-module L ( λ ) a K − µ -eigenvector witheigenvalue (cid:54) = 1, say the highest weight vector. But B contains a copy of U q ( sl )and all one-dimensional representations of this require K − µ to act = 1. Thus therestriction of L ( λ ) to B has to have some higher-dimensional irreducible compo-sition factor (note that the eigenvalue argument does not require semisimplicity).Hence B is in this case not basic. (cid:3) Induction of one-dimensional characters
Definition and first properties.
One of the reasons Borel subalgebras ofa Lie algebra or quantum group are interesting, is because they can be used toconstruct induced modules and a category O . An interesting implication of definingand classifying unfamiliar Borel subalgebras B is to study the respective inducedrepresentations from B to H = U q ( g ) of any one-dimensional B -module k χ : V ( B, χ ) := U q ( g ) ⊗ B k χ As defining property for Borel algebras B we chose to generalize the Lie algebraterm “maximal solvable” by “maximal with the property basic ” i.e. all irreduciblefinite-dimensional representations are one-dimensional. This matches the upcom-ing purpose, since we can again use solely one-dimensional characters of B . For aright coideal subalgebra B the category Rep( B ) is not a tensor category, but a rightRep( H )-module category via M ⊗ k V with B -action given by b (1) ⊗ b (2) ∈ C ⊗ H . Lemma 4.1.
For a right coideal subalgebra of a Hopf algebra B ⊂ H , the inductionfunctor Rep( B ) → Rep( H ) and the restriction functor Rep( H ) → Rep( B ) are bothmorphisms of right Rep( H ) -module categories.Proof. Restriction is a morphism of module categories with trivial structure mapRes( V ⊗ W ) = Res( V ) ⊗ W due to the right coideal subalgebra property. Simi-larly, Induction is a morphism of module categories with structure map given by( H ⊗ B M ) ⊗ V ∼ = H ⊗ B ( M ⊗ V ) h ⊗ m ⊗ v (cid:55)→ h (1) ⊗ m ⊗ S ( h (2) ) .v which is easily checked to be an H -module morphism. (cid:3) Remark 4.2.
In our examples H is free as a B module, which is very helpfulin constructing the induced representations. This does not follow directly from thecelebrated 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 ourcase. We feel that B being free over U is the decisive property. Our main interest is how induced representations of Borel subalgebras decom-pose as U q ( g )-modules. For a first result, we use part of our defining property: Lemma 4.3.
Let H be a Hopf algebra and B ⊂ H a basic right coideal subalgebra.Then any finite-dimensional irreducible H -module V is a quotient of an inducedmodule H ⊗ B k χ for some character χ .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 inparticular there is a nontrivial B -module monomorphism k χ → V for a suitable χ .This induces up to a nonzero H -module morphism H ⊗ B k χ → V . For V irreduciblethis morphism has to be surjective. (cid:3) Example 4.4.
Let B = U ≤ be the standard Borel subalgebra of U q ( g ) , then everyone-dimensional character χ : B → k is zero on U − and thus comes from somegroup character χ : Λ → k × . The induced module U ⊗ B k χ is isomorphic to U + as a U -module. It is a highest-weight module generated by the vector v with K µ .v = χ ( K µ ) v and E α .v = 0 . We shall see that also for our new Borel subalgebras B the induced modules forgeneric χ are irreducible, while special values of χ have as quotient each of theirreducible finite-dimensional U q ( g )-modules L ( λ ). However the action of U willnot be diagonalizable any more. Problem 4.5.
Is it true that any induced module from a Borel subalgebra have aunique irreducible quotient? It is to expect that maximality enters here.
Problem 4.6.
Can one use the graded algebra in the upcoming Conjecture Ato understand these modules and determine their decomposition behaviour of theinduced modules? We would expect that gr( B ) plays a similar role as U in thestandard case, so the graded modules gr( V ) should be gr( B ) -diagonalized. Problem 4.7.
Can one define a category O with nice properties? In particular itshould be an abelian category, closed under submodules and quotients, with enoughprojectives and injectives. Example A . We now want to construct and decompose all induced repre-sentations in the case sl . From Section 3.2 we know all Borel algebras of U q ( g ) up to reflection:In the familiar case B std = U ≤ = (cid:104) K, F (cid:105) all one-dimensional representationsare of the form χ ( F ) = 0 and χ ( K ) arbitrary. Then the induced module is V ( B std , χ ) := U q ( g ) ⊗ B k χ ∼ = k [ E ]1 χ with highest weight vector K. χ = χ ( K )1 χ . All of these modules have a diagonalaction of K . For certain integral choices V ( B std , χ ) has a finite-dimensional irre-ducible quotient and all finite-dimensional irreducible U q ( g )-representations arisethis way.The novel case in Example 3.6 is B λ,λ (cid:48) := (cid:104) ¯ E, ¯ F (cid:105) with ¯ E := EK − + λK − , ¯ F = F + λ (cid:48) K − and arbitrary scalars λλ (cid:48) = q (1 − q )( q − q − ) , an algebra isomorphic to thequantized Weyl algebra. As we showed, all finite-dimensional irreducible repre-sentations factorize over the commutative quotient (cid:104) ¯ E, ¯ F (cid:105) / ( ¯ E ¯ F − q ( q − q − )(1 − q ) ).Hence all one-dimensional representations are of the form χ ( ¯ E ) = e , χ ( ¯ F ) = f with ef = q ( q − q − )(1 − q ) = λλ (cid:48) . Using the PBW basis we easily get V ( B λ,λ (cid:48) , χ ) := U q ( g ) ⊗ B λ,λ (cid:48) k χ ∼ = k [ K, K − ]1 χ and we calculate the action to be K.K n χ = K n +1 χ F.K n χ = q n K n F χ = q n f · K n χ − q n λ (cid:48) · K n − χ E.K n χ = q − n − K n +1 EK − χ = q − n − e · K n +1 χ − q − n − λ · K n χ or in matrices K = · · ·
01 01 0 · · · F = · · · q n − f − q − n λ (cid:48) q n f − q − n +1) λ (cid:48) q n +1) f · · · E = · · · − q − n − − λq − n − − e − q − n − λq − n − e − q − n +1) − λ · · · In particular, the action of K is not diagonalizable. Moreover the action of K shows that no V ( B λ,λ (cid:48) , χ ) has finite-dimensional proper submodules.We can determine all submodules with a trick that does not seem to easilygeneralize beyond rank one: Lemma 4.8.
The induced U q ( sl ) -module with respect to a Borel algebra B λ,λ (cid:48) V ( B λ,λ (cid:48) , χ ) := U q ( g ) ⊗ B λ,λ (cid:48) k χ ∼ = k [ K, K − ]1 χ has a nontrivial submodule V (cid:48) iff for some n ∈ N χ ( ¯ E ) = (cid:15)q n · λ, equivalently χ ( ¯ F ) = (cid:15)q − n · λ (cid:48) It is cofinite of codimension [ V : V (cid:48) ] = n + 1 .Proof. A U q ( sl )-submodule W ⊂ V ( B λ,λ (cid:48) , χ ) ∼ = k [ K, K − ] is in particular a k [ K, K − ]-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 ) = (cid:80) finiten ∈ Z 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:( F.P )( X ) = (cid:88) n c n q n f · X n − q n λ (cid:48) · X n − = P ( q X ) (cid:0) f − λ (cid:48) X − (cid:1) ( E.P )( X ) = (cid:88) n c n q − n − e · X n +1 − q − n − λ · X n = P ( q − X ) q − ( eX − λ )For degree reasons this can only give a multiple of P if for some a, b, c, d :( aX + b ) P ( X ) ! = P ( q X ) ( f X − λ (cid:48) )( cX + d ) P ( X ) ! = P ( q − X ) q − ( eX − λ )Either P ( X ) is constant, then W is the entire module, or the zeroes of P have tolay in a chain q − n X , q − n +2 X , . . . q n X for some n ∈ N , with q n X = λ (cid:48) /f and q − n X = λ/e . The quotient module is thus the ( n + 1)-dimensional ring extension k [ K, K − ] / ( P ), where K acts again by left-multiplication. (cid:3) We show a different and quite general idea to detect the finite-dimensional quo-tient modules (i.e. the cofinite submodules) and circumvents the problem of non-diagonalizable K -action: To find finite-dimensional quotient modules we need tofind elements in Hom U q ( g ) ( U q ( g ) ⊗ B k χ ) and this can be done by decomposingHom k ( U q ( g ) , k ) into irreducible L ( λ ). This is now possible since Hom k ( U , k ) is di-agonalizable 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 ( sl ) -module with respect to a Borel algebra B λ,λ (cid:48) V ( B λ,λ (cid:48) , χ ) := U q ( g ) ⊗ B λ,λ (cid:48) k χ ∼ = k [ K, K − ]1 χ has a finite-dimensional quotient the irreducible module L ( (cid:15), n ) of dimension n + 1 and sign (cid:15) = ± for precisely one choice of χ , namely χ ( ¯ E ) = (cid:15)q n · λ, equivalently χ ( ¯ F ) = (cid:15)q − n · λ (cid:48) Proof.
Let φ : V ( B λ,λ (cid:48) , χ ) → L ( (cid:15), n ) be a (nonzero) module homomorphism to theirreducible highest weight module with highest weight vector v for highest weight Kv = q n v . It has a basis v , . . . v n with Kv i = (cid:15)q n − i v i . Denote the coefficients of φ in this basis by φ k , k = 0 . . . n ,. Then by definition (cid:15)q n − k φ k ( K i χ ) = K.φ k ( K i χ ) = φ k ( K i +1 χ )Thus φ is fixed by its image φ k (1 χ ) via φ k ( K i χ ) = (cid:15) i q ( n − k ) i φ k (1 χ )and since the map should be surjective we need all φ k (1 χ ) (cid:54) = 0 for k = 0 . . . n . On the other hand the action of F and E demands: φ k − ( K i χ ) = φ k ( F.K i χ )= q i f φ k ( K i χ ) − q i λ (cid:48) χ k ( K i − χ )= q i (cid:0) (cid:15) i q ( n − k ) i f − (cid:15) i − q ( n − k )( i − λ (cid:48) (cid:1) φ k (1 χ ) (cid:15) [ k + 1] q [ n − k ] q φ k +1 ( K i χ ) = φ k ( E.K i χ )= q − i − eφ k ( K i +1 χ ) − q − i − λφ k ( K i χ )= q − i − (cid:0) (cid:15) i +1 q ( n − k )( i +1) e − (cid:15) i q ( n − k ) i λ (cid:48) (cid:1) φ k (1 χ )and the boundary conditions of the first resp. second equation for k = − k = n that the right-hand side has to be zero0 = (cid:15)f − q − n λ (cid:48) (cid:15)q − n e − λ which is the asserted condition. One could also derive direct formulae for φ k ( K i χ )and in this case it is easy to see that φ above is indeed a module homomorphism. (cid:3) The structure of the graded algebra of a right coidealsubalgebra
Conjecture A.
By Theorem 2.5, right coideal subalgebras of U − q ( g ) are char-acter shifts k [ L ] U − [ w ] φ with L ⊂ Λ orthogonal to supp( φ ). These character-shiftsare not Λ-graded. An important structural insight into these algebras, and subse-quently right coideal subalgebras of U q ( g ) and there representation theory, wouldbe the knowledge of the respective graded algebras. Take the Z -grading of U q ( g )where deg( E α i ) = 1, deg( K α i ) = 0, deg( F α i ) = −
1. then we conjecture:
Conjecture A.
Fix some w ∈ W and character φ and consider the canonical map f : gr( U − [ w ] φ ) → U ≤ sending an element to the summands of leading degree.Then the image D of f is a Z -graded right coideal subalgebra of U ≤ and has thefollowing properties: • D is the semigroupring of G ( D ) = { K − µ | µ ∈ supp( φ ) } . We denote by ˜ G its quotient group or localization. • Then for the explicit element w (cid:48) = ( (cid:81) β ∈ supp( φ ) s β ) w we have k [ ˜ G ( D )] D = k [ ˜ G ( D )] U − [ w (cid:48) ] . Morally , this assertion means gr( U − [ w ] φ ) = U − [ w (cid:48) ] up to localization, and themain part of the assertion is the explicit formula of w (cid:48) . The first step of the proof of Conjecture A is to observe that D is a rightcoideal subalgebra of U − q ( g ). If D = k [ Z k ] is a full group ring, then Theorem2.3 asserts that that D ∼ = U − [ w (cid:48) ] for some w (cid:48) ∈ W . However, D need not be a group so there are many possible (closely related) D generated by linearcombinations of character-shifted E α , F α with different K -prefactors, see Example5.5. To circumvent these problems, we localize G to ˜ G , then Theorem 2.3 asserts k [ ˜ G ( D )] D = k [ ˜ G ( D )] U − [ w (cid:48) ]. Our task is to prove the asserted formula for w (cid:48) ,which captures the information which E α , F α 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 givesome examples and formulate the consequence for triangular coideal subalgebras. Example 5.1.
Clearly for φ = 0 the right coideal subalgebra U − [ w ] φ is alreadygraded and w (cid:48) = w . Example 5.2.
Take our example B λ,λ (cid:48) with negative part U − [ s α ] φ − = (cid:104) F + λ (cid:48) K − (cid:105) for φ − ( F ) = λ (cid:48) (cid:54) = 0 . Then the graded algebra is as conjectured with w = s α , w (cid:48) = 1 : gr( B − λ,λ (cid:48) ) = U − [1] k [ K − ] = (cid:104) K − (cid:105) Altogether, the graded algebra of B λ,λ (cid:48) is the positive Borel part, up to localization: gr( B λ,λ (cid:48) ) = U − [1] k [ K − ] S ( U + [ s α ] = (cid:104) K − , KE (cid:105) Example 5.3.
Let g = sl , w = w = s s s and χ ( F ) = λ (cid:48) , supp( χ ) = { α } .Then U − [ w ] χ = (cid:104) F + λ (cid:48) K − , F (cid:105) . The leading degree terms of the character-shift ¯ F = F + λ (cid:48) K − is λ (cid:48) K − , and ¯ F = F . Typically, character-shifts of elements of Λ -degree − ( α + α ) have leading degree terms in Λ -degree − α : F F = F F + λ (cid:48) K − · F F F = F F + F · λ (cid:48) K − However, in a suitable linear combination of these two elements, the leading degreeterms of the character-shifts cancel and we again land in Λ -degree − ( α + α ) : F F − q − F F = F F − q − F F Thus the graded algebra is as conjectured, with w (cid:48) = s w = s s : f : gr U − [ s s s ] χ ∼ = U − [ s s ] k [ K − ] We remark that the definition of a root vector depends on the choice of a reducedexpression for w , while the linear combination we found above does not, so it canbe a root vector or not. In contrast, for a right coideal subalgebra in the positive part U S ( U + ) thegraded right coideal subalgebra has the same degrees, i.e. by choice of our gradingthe leading terms of the character-shifted ¯ E α are simply the E α themselves. f : gr( U + [ w ] φ ) ∼ −→ U + [ w ]The two statements can be easily combined: Corollary 5.4.
Assuming Conjecture A holds for U q [ w − ] φ − i.e. for D := gr( U q [ w − ] φ − ) holds after completing the semigroup G ( D ) to the quotient group ˜ G ( D ) : k [ ˜ G ( D )] gr( U q [ w − ] φ − ) ∼ = k [ ˜ G ( D )] U − [ w (cid:48) ] Then for any triangular right coideal subalgebra the map sending elements to theirleading degree terms gives an isomorphism of right coideal subalgebras gr (cid:0) U − [ w − ] φ − k [ L ] S ( U + [ w + ]) φ + (cid:1) ∼ = D k [ L ] S ( U + [ w + ]) k [ ˜ G ( D )] gr (cid:0) U − [ w − ] φ − k [ L ] S ( U + [ w + ]) φ + (cid:1) ∼ = U − [ w (cid:48) ] k [ ˜ G ( D ) L ] S ( U + [ w + ])As the representation theory of the graded algebra is well understood, we wouldultimately hope that this puts explicit and sharp condition when a choice of( w + , φ + , w − , φ − ) 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 technicaldifficulty, which we address in this section is that in general A (cid:48) = gr( A ) hasin general A (cid:48) = k [ G ] for a semigroup G ⊂ Z k . Such a Z -graded right coidealsubalgebra may not even be N k -graded. Differently spoken, it does not have to bea free module over the semigroup G . Example 5.5.
Consider for example the subalgebra in U q ( sl ) − generated as: C = (cid:104) F K − + F K − , K − , K − (cid:105) It is closed under comultiplication and thus a right coideal subalgebra.Multiplication with K − or K − from either side commute differently with F , F ,which shows that the following are elements in CC (cid:51) F K − K − , F K − K − but neither F , F are in C . These Z -graded right coideal subalgebras A (cid:48) are not covered by the classificationTheorem 2.3. We shall not try to classify A (cid:48) directly, but study its localization: Proposition 5.6.
Let A (cid:48) ⊂ U − be a Z -graded right coideal subalgebra, and definethe semigroup G by A (cid:48) = A (cid:48) ∩ U = k [ G ] , then k GA (cid:48) = A (cid:48) k G and thus we candefine a localization in the noncommutative setting [Sten75] Prop 1.4:Let ˜ G ⊂ Z k be the quotient group (localization) of G . Then ˜ A (cid:48) := k [ ˜ G ] A (cid:48) = A (cid:48) k [ ˜ G ] is again a right coideal subalgebra and there exists a w (cid:48) ∈ W with ˜ A (cid:48) = k [ ˜ G ] U − [ w (cid:48) ] Proof.
Since A (cid:48) is an algebra we have KA (cid:48) = A (cid:48) K for K ∈ G , and by invertibility of K we have A (cid:48) K − = K − A (cid:48) K , which implies the first claim. Now, ˜ A (cid:48) is Z -graded, so˜ A (cid:48) = ˜ A (cid:48)≤ ˜ A (cid:48)≥ , and after localization both factors fulfills the conditions of Theorem Z -graded, the character-shifthas to be trivial and hence ˜ A (cid:48) is as asserted, in particular Z n -graded. (cid:3) Proposition 5.7.
Let G be a semigroup with ˜ G its quotient group. Let ˜ V be a ˜ G -module and V be a G -submodule, such that ˜ GV = ˜ V . Then for any v ∈ ˜ V thereexists a K ∈ G such that Kv ∈ V .Proof. Since we assumed ˜ GV = ˜ V , there exist finitely many elements K ( i ) ∈ ˜ G and vectors v ( i ) ∈ V such that n (cid:88) i =1 K ( i ) v ( i ) = v Now by definition of ˜ G there exists for each element K ( i ) an element K ( i ) ∈ G with K ( i ) K ( i ) ∈ G . Taking some K ∈ G in (cid:84) i K ( i ) G (for example their product) provesthe assertion. (cid:3) Applying this to k [ ˜ G ] A (cid:48) = k [ ˜ G ] U − [ w (cid:48) ] we get the following easy consequence,which will also be useful later-on: Corollary 5.8.
For any root µ ∈ Φ( w (cid:48) ) and a reduced expression for w (cid:48) , thereexists a K ∈ G , such that KF µ ∈ A (cid:48) . Next, we define a suitable growth-condition that replaces the length (cid:96) ( w (cid:48) ) re-spectively the number of PBW generators for arbitrary non-homogeneous A (cid:48) : Definition 5.9.
Let V be an N -graded vector space with finite-dimensional homo-geneous components V n . Let H ( V, z ) = (cid:80) n ≥ dim( V n ) z n be the Hilbert series, thenwe define growth( V ) = a such that < lim inf z →∞ (1 − z ) a H ( V, z ) and lim sup x →∞ (1 − z ) a H ( V, z ) < + ∞ otherwise we say the growth is undefined. The definition is made in a way, such that V sub ⊂ V ⊂ V sup with well-defined growth( V sub ) = growth( V sup ) =: a implies an intermediate well-definedgrowth( V ) = a . Note that H ( V, z ) needs not be a rational function, even if H ( V sub , z ) , 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 ingenerators x , . . . x (cid:96) of degree d , . . . d (cid:96) ≥ , then growth( V ) = (cid:96) , since explicitly H ( V, z ) = (cid:96) (cid:89) i =1 − z d i When we apply this notion and result to our
A, A (cid:48) ⊂ U − , we need to con-sider a modified grading , which has finite-dimensional homogeneous components,but equal behaviour with respect to character shifts: Define the modified degreemdeg( − ) = deg( − ) + sdeg( − ) by adding a second degree sdeg( F i ) = sdeg( K ± i ) = −
1, so mdeg( F i ) = − K ± i ) = −
1. Note that the algebra is not gradedany more. Clearly the homogeneous components of mdeg( − ) on U − are finite-dimensional. Since characters shifts are graded with respect to sdeg( − ), this newdefinition does not change the vector space gr( A ):gr deg( − ) ( A ) = gr mdeg( − ) ( A ) Example 5.11.
Let A = U − [ w ] with the modified grading, then growth( A ) = (cid:96) ( w ) .More generally, let G be a semigroup with ˜ G = Z k and take A (cid:48) = k [ G ] U − [ w (cid:48) ] , then growth k ( A ) = rank ˜ G + (cid:96) ( w (cid:48) ) . We now need to compare the growth of A = U − [ w ] φ , or equivalently A (cid:48) = gr( A ),to its localization k [ ˜ G ] A (cid:48) = k [ ˜ G ] U − [ w (cid:48) ]: Lemma 5.12.
Let V be a N -graded module over a semigroup G ⊂ Z k . Then wehave for the localization We have for the localization growth( V ) = growth( k [ ˜ G ] V ) .Proof. Let ˜ V = ˜ GV , which is a free k [ ˜ G ]-module. Let I be a ˜ G -basis of ˜ V , thenby Proposition 5.7 there are elements K ( i ) ∈ G with K ( i ) v i ∈ V and we define itsspan to be V sub ⊂ V . On the other hand we may consider V sup = ˜ V ⊃ V .Then we have an inequality by definitiongrowth(V sub ) ≤ growth(V) ≤ growth(V sup )but in our case growth(V sub ) = growth(V sup ), which proves the assertion (cid:3) Corollary 5.13.
Let now again A := U − [ w ] φ and A (cid:48) := gr( A ) and the localization k [ ˜ G ] A (cid:48) = k [ ˜ G ] U − [ w (cid:48) ] . Then comparing the growth gives (cid:96) ( w ) = rank( ˜ G ) + (cid:96) ( w (cid:48) )5.3. Induction step.
Let w ∈ W and φ be a character on U − [ w ] with supportsupp( φ ) ⊂ Φ + ( w ), a set of orthogonal simple roots. We continue to study thecharacter-shifted coideal subalgebra, its graded algebra and its localization A := U − [ w ] φ , A (cid:48) := gr( A ) , ˜ A (cid:48) = k [ ˜ G ] A (cid:48) = k [ ˜ G ] U − [ w (cid:48) ]for some w (cid:48) ∈ W , whose explicit description is the goal of Conjecture A. Weattempt an induction on the length (cid:96) ( w ). Lemma 5.14.
Assume Conjecture A holds for all u ∈ W with (cid:96) ( u ) < (cid:96) ( w ) andall characters. Assume w = us α i with (cid:96) ( w ) = (cid:96) ( u ) + 1 for some α i . Then one ofthe following happens: • If u ( α i ) ∈ supp( φ ) then Theorem 5.1 holds for w and w (cid:48) = u (cid:48) , which is the w (cid:48) asserted in this case. • If u ( α i ) ∈ supp( φ ) then there exists a simple root α j such that w (cid:48) := u (cid:48) s α j has length (cid:96) ( u (cid:48) ) + 1 and k [ ˜ G ( A (cid:48) )] A (cid:48) = k [ ˜ G ( A (cid:48) )] U − [ w (cid:48) ] Conjecture A holds for w iff α j = α i . Proof.
By [HK11b] p. 13 for u < w the restriction φ u of φ to U − [ u ] ⊂ U − [ w ]is again a character with supp( φ u ) ⊆ supp( φ ). The inductive assumption thatTheorem 5.1 holds for U − [ u ] φ u means that we have A (cid:48) u := gr (cid:0) U − [ u ] φ u (cid:1) A (cid:48) u = k [ (cid:104) K − µ , µ ∈ supp( φ u ) (cid:105) ] k [ ˜ G u ] A u ∼ = k [ ˜ G u ] U − [ u (cid:48) ]where u (cid:48) has the asserted form u (cid:48) := ( (cid:81) β ∈ supp( φ u ) s β ) u , and where ˜ G u is the quo-tient group of the semigroup G u generated by supp( φ u ), with A (cid:48) u = k [ G u ].Now for the new element w ∈ W with Φ + ( w ) = Φ + ( u ) ∪{ u ( α i ) } a new character-shifted root vector ¯ F u ( α i ) appears in the PBW-basis A := U − [ w ] φ = U − [ u ] φ u (cid:10) F u ( α i ) (cid:11) A (cid:48) = gr (cid:0) U − [ w ] φ (cid:1) We now apply Corollary 5.13 which asserts (cid:96) ( w ) = rank( ˜ G ) + (cid:96) ( w (cid:48) ) (cid:96) ( u ) = rank( ˜ G u ) + (cid:96) ( u (cid:48) )We have (cid:96) ( w ) = (cid:96) ( u ) + 1. Moreover by inclusion A (cid:48) ⊇ A (cid:48) u and Φ + ( w (cid:48) ) ⊇ Φ( u (cid:48) )we have inequalities rank( ˜ G ) ≥ rank( ˜ G u ) and (cid:96) ( w (cid:48) ) ≥ (cid:96) ( u (cid:48) ). Hence we have twopossible cases:i) rank( ˜ G ) = rank( ˜ G u ) + 1 and (cid:96) ( w (cid:48) ) = (cid:96) ( u (cid:48) ). The latter implies w (cid:48) = u (cid:48) ,the former implies rank( ˜ G ) (cid:13) rank( ˜ G u ). Hence the leading term of ¯ F u ( α i ) must be the new element K − µ ∈ A (cid:48) . By orthogonality of supp( φ ), we findthat µ = u ( α i ) is a new simple root α n in supp( φ ) = { µ } ∪ supp( φ u ). Inparticular w = us α i = s α n u . This proves that w (cid:48) = u (cid:48) is as asserted byConjecture A as( (cid:89) β ∈ supp( φ ) s β ) w = s α n u (cid:48) s α i = u (cid:48) = w (cid:48) ii) rank( ˜ G ) = rank( ˜ G u ) and (cid:96) ( w (cid:48) ) = (cid:96) ( u (cid:48) ) + 1. The former implies ˜ G = ˜ G u andsupp( φ ) = supp( φ u ). The second implies together with Φ + ( w (cid:48) ) ⊃ Φ + ( u (cid:48) )that there exists a simple root α j with w (cid:48) = u (cid:48) s α j . Since the support isunchanged, the assertion of Conjecture A would be α j = α i . (cid:3) The second case is what we need to study in what follows. We remark that thenew element in the graded algebra in degree u (cid:48) ( α j ) ∈ Φ + ( w (cid:48) ) does not have tobe directly the leading term of the character-shift F u ( α i ) . We illustrate this in anexplicit example: Example 5.15.
Let g = A and u = s s s , then Φ + ( u ) = { β , β , β } supp = (cid:104) u ( − γ ) , u ( − γ ) (cid:105) = (cid:104) β , β (cid:105) The algebra U − [ s s s ] has a PBW basis in the generators F , F = T − F , F := T − T − F = [ F , [ F , F ] q +1 ] q +1 In accordance with the previous theorem, the character-shifted root vectors ¯ F = F + φ ( F ) K − ¯ F = F + φ ( F ) K − ¯ F = F + φ ( F )[ K − , [ F , F ] q ] q + φ ( F )[ F , [ K − , F ] q ] q + φ ( F ) φ ( F )[ K − , [ K − , F ] q ] q = F + φ ( F )( q − − q )[ F , F ] q K − + φ ( F )( q − − q )[ F , F ] q K − + φ ( F ) φ ( F )( q − − q ) F K − K − have highest degrees { , , β } and thus the graded algebra corresponds to the Weylgroup element u (cid:48) = s s u = s as conjectured: gr (cid:0) U − [ s s s ] φ (cid:1) ∼ = gr (cid:0) U − [ s ] (cid:1) (cid:104) K − , K − (cid:105) Now the critical induction step is to prolong this by α i = β to w = us , thenwe have a new root vector in degree u ( α i ) = β , to be precise: F := T − T − T − F = T − F = [ F , F ] q +1 ¯ F = [ F , F ] q + φ ( F )[ K − , F ] q = F + φ ( F )( q − − q ) F K − The leading degree term F of ¯ F is already contained in U − [ u (cid:48) ] , as leading degreeterm of ¯ F . The new degree u (cid:48) ( α j ) in U − [ w (cid:48) ] ⊃ U − [ u (cid:48) ] will be the leading degreeof some suitable linear combination, and the main question is which will be thenext-leading degree. Simply from degree considerations there are two reasonablepossibilities (in the sense of criterion 1 below): • u ( α i ) = u (cid:48) ( α j ) = β , contrary to Conjecture A.Actually this seems the natural choice, because it is the next leading degreein ¯ F . • u ( α i ) + β − β = u (cid:48) ( α i ) = β , according to Conjecture A.For this to happen, F , F have to cancel simultaneously in a suitablelinear combination of ¯ F , ¯ F , and then F from ¯ F is the leading term.Having explicit expressions for ¯ F , ¯ F we can check explicitly that the latter isthe case, because the coefficients of F , F in ¯ F , ¯ F are proportional: ¯ F − φ ( F )( q − q − ) ¯ F K − = F + φ ( F )( q − − q )[ F , F ] q K − We note that already the non-vanishing of the F -term would have sufficed inthis situation, because neither β , β are larger then the other and we know fromlength that we only get a single new degree. Three Criteria implying Conjecture A.
Let C = U − [ w ] φ with w = us α i , u ( α i ) (cid:54) = supp( φ ) and let gr( U − [ u ] φ ) = U − [ u (cid:48) ] for some u (cid:48) , then the secondcase in Lemma 5.14 states that gr( U − [ w ] φ ) = U − [ w (cid:48) ] for w (cid:48) = u (cid:48) s α j . Conjecture Ais concerned with the shape of u (cid:48) , w (cid:48) , and it would hold inductively if α j = α i .We now collect three criteria when this is the case. Each criterion by itself showsthe induction step for certain cases w = us α i . Besides they give the possibility tocheck it manually in a given case - this is in particular interesting for Criterion 3.As first criterion we note the general relation between α i and (the potentiallydifferent) α j , which allows us to classify critical induction steps up to reflection. Lemma 5.16 (Criterion 1) . a) We have the necessary criterion u ( α i ) − u (cid:48) ( α j ) ∈ Z [supp] .b) Assume (contrary to Conjecture A) that α i (cid:54) = α j for u ( α i ) − u (cid:48) ( α j ) ∈ Z [ S ] with S ⊂ supp a subset of S , which we choose minimal. Then u ( α i ) , u (cid:48) ( α j ) , S are upto reflection in a parabolic subsystems Φ r ⊂ Φ of rank , as follows: Φ r u ( α i ) − u (cid:48) ( α j ) ∈ Z [ S ] A α − α = α + α B α − α = α + α B α − α = α + α + α C α − α = α + α D α − α = α + α + α or u ( α i ) = u (cid:48) ( α j ) . If Conjecture A holds already for u , then we have in addition u ( α i ) − u ( α j ) ∈ Z [ S ] , so also this tuple is in the list above.Proof. a) The proof of Lemma 5.14 shows that U − [ w ] φ contains a new element¯ F u ( α i ) and since φ is a character-shifts with respect to supp, this algebra is stillgraded by cosets of Z [supp]. On the other hand the graded algebra A (cid:48) containsa new element in degree u (cid:48) ( α j ).b) Any set of k roots can be reflected to a parabolic subsystem of rank at most k .Applying this assertion to S, β i gives a subsystem Φ r of rank r = | S | + 1, whichis moreover connected (otherwise the equation would hold even for a subset S (cid:48) smaller S ). Applying the assertion a second times to S gives a subset of | S | = r − | S | >
1, this can only be truefor Φ r = A , B , C with S reflected to the outmost simple roots α , α , or forΦ r = D with S reflected to the three outmost roots α , α , α . The result thenfollows from directly inspecting these root systems for solutions β i − β j ∈ Z [ S ]. (cid:3) As a second criterion, we formulate the consequence of applying the inductionstep to two different presentations of w : Definition 5.17.
A Weyl group element w has a unique ending iff one of thefollowing equivalent conditions applies: • Any reduced expression of w ends in the same letter s α i . • Φ + ( w − ) contains a unique simple root α i . • w ( α k ) > for all simple roots α k expect for one α i . Lemma 5.18 (Criterium 2) . Assume that w has not a unique ending, i.e. w = u s α i = u s α l with α i (cid:54) = α l , and u ( α i ) , u ( α l ) (cid:54)∈ supp . Then if Conjecture A holdsfor u , u , then α j = α 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 movesto transform the two presentations of w into one another. In particular under ourassumption there exists a reduced expression of w which ends w = r · · · s α i s α l s α i = r · · · s α i s α l s α i of length 2 , , , α l , α i ) , i (cid:54) = l , such that a braid group move canbe performed. Differently spoken u := r · · · s α i s α l , w = u s α i u := r · · · s α l s α i , w = u s α l Assume that Conjecture A holds for the elements u , u of shorter length, thenLemma 5.14 asserts that there exists α j , α k with w (cid:48) = u (cid:48) s α j = u (cid:48) s α k and in particular Φ( u (cid:48) ) , Φ( u (cid:48) ) ⊂ Φ( w (cid:48) )In case we have α i (cid:54) = α j , this would imply already Φ( u (cid:48) ) ⊂ Φ( u (cid:48) ), which is acontradiction. (cid:3) As a third criterion we prove the assertion about U − [ w ] for w = us α i by directcalculation. In general, it is not easy to relate the coproduct of a reflected elementto the coproduct of the original element, but we can obtain enough informationunder a multiplicity-zero condition. It enters through the following formulae: Proposition 5.19.
Let α m be a simple root and consider T − v − ( F i ) for any v ∈ W, i ∈ I , which is in degree ν := v ( α i ) . Assume that ( α m , ν ) = − and that α m has multiplicity zero in the degree of v ( α i ) . Then ( T − s m T − v − ( F i )) φ m = T − s m T − v − ( F i ) + ( q m − q − m ) φ m ( F m ) · T − v − ( F i ) K − m If ( α m , ν ) = − c a similar formula holds. Proof.
To compute the character-shift ( T s m T − v − ( F i )) φ m we need to relate coproduct andreflection, which is involved. From [Lusz93] Prop 37.3.2 we have the formula ∗ ( T − s m ⊗ T − s m )∆( T s m ( x )) = (cid:32) ∞ (cid:88) l =0 q l ( l − / m { l } q m F ( l ) m ⊗ E ( l ) m (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) L (cid:48) ∆( x ) (cid:32) ∞ (cid:88) r =0 ( − r q − r ( r − / m { r } q m F ( r ) m ⊗ E ( r ) m (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) L (cid:48)(cid:48) where { l } q m = [ l ] q m ( q m − q − m ) l . We need an analogous formula for T − s m , by substi-tuting x = T − s m ( y ) and using L (cid:48) L (cid:48)(cid:48) = 1 from [Lusz93] Prop. 5.3.2∆( T − s m ( y )) = (cid:32) ∞ (cid:88) r =0 ( − r q − r ( r − / m { r } q m F ( r ) m ⊗ E ( r ) m (cid:33) ( T − s m ⊗ T − s m )∆( y ) (cid:32) ∞ (cid:88) l =0 q l ( l − / m { l } q m F ( l ) m ⊗ E ( l ) m (cid:33) If we apply the character φ m ⊗
1, then the only nonzero contributions are thosetensor summands of ( T − s m ⊗ T − s m )∆( y ) and hence ∆( y ) where the left tensor factorhas degree in Z α m .We now use our assumption that y := T − v − ( F i ) in degree v ( α i ) does not containa factor E m . Thus in this case the only nonzero contribution comes from thesummand 1 ⊗ T − s m T − v − ( F i ) in ( T − s m ⊗ T − s m )∆( y ), thus in this case( φ m ⊗ T − s m ( y )) = ∞ (cid:88) r,l =0 ( − r q − r ( r − / m { r } q m q l ( l − / m { l } q m · φ m ( F ( r ) m F ( l ) m ) · E ( r ) m (cid:0) T − s m T − v − ( F i ) (cid:1) E ( l ) m = ∞ (cid:88) s =0 q s ( s − / m ( q m − q − m ) s φ m ( F m ) s s (cid:88) r =0 ( − r q (1 − s ) rm · E ( r ) m (cid:0) T − s m T − v − ( F i ) (cid:1) E ( s − r ) m By reasons of degree (or by the quantum Serre relation), all terms s (cid:54) = 0 , α m , ν ) = −
1. If ( α m , ν ) = − c , then similarly only the terms up to s = c arenonzero.The last equality can then by most easily calculated by hand, since by theassumption on the multiplicity of α m , we have that T v ( F i ) is a linear combinationsof terms F (cid:48) F k F (cid:48)(cid:48) where F (cid:48) , F (cid:48)(cid:48) are products of F l with ( α m , α l ) = 0. Then by themultiplicativity of T − s m and the defining formulae [Jan96] Sec 8.14 (8’) we have( T − s m T − v − ( F i ) = T − s m ( F (cid:48) F k F (cid:48)(cid:48) ) = F (cid:48) ( T − s m ( F k )) F (cid:48)(cid:48) = F (cid:48) [ F m , F k ] q m F (cid:48)(cid:48) and similarly by [ E m , F l ] = [ K m , F l ] = 0 we get further[ E m , ( T − s m T − v − ( F i )] = F (cid:48) [ E m , [ F m , F k ] q m ] F (cid:48)(cid:48) = F (cid:48) F k K − m F (cid:48)(cid:48) = F (cid:48) F k F (cid:48)(cid:48) K − m ∗ We use the notation T s m , T − s m in [Jan96], which translates by the remark on p. 146 toLusztig’s notation T (cid:48)(cid:48) m, , T (cid:48) m, − . This proves the assertion. (cid:3)
Lemma 5.20 (Criterion 3) . Let w = us α be a Weyl group element u = s m s n u (cid:48) and a character φ with support containing { α m , α n } . Assume that the simple root α m appears in u ( α i ) − α m ∈ Φ 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 (cid:48) s i ] which has acharacter-shift with leading term T − u (cid:48)− ( F i ) as asserted. , using the inductive as-sumption on u = s m s n u (cid:48) as well as on s n u (cid:48) s i and on s m u (cid:48) s i for smaller support { α n } resp. { α m } . For the third inductive assumption we require the stronger ex-plicit assertion from Proposition 5.19 for v = u (cid:48) that relies on the assumption ofmultiplicity one:( T − s m T − u (cid:48)− ( F i )) φ m = T − s m T − u (cid:48)− ( F i ) + ( q m − q − m ) φ m ( F m ) · T − u (cid:48)− ( F i ) K − m Now the proof proceeds as follows: By the inductive assumption on s n u (cid:48) s i thereexists an X ∈ U − [ s n u (cid:48) s i ] such that we have X φ n = T − u (cid:48)− ( F i ) in gr U − [ s n u (cid:48) s i ] φ n , ordifferently spoken the character-shift has leading term X φ n = T − u (cid:48)− ( F i ) + · · · We may assume X to be chosen in degrees u (cid:48) ( α i ) + Z α n , because U − [ s n u (cid:48) s i ] φ n isstill graded by Z α n -cosets.We now apply T − s m , which commutes with the φ n character-shift, so we have con-structed an element T − s m ( X ) ∈ U − [ s m s n u (cid:48) s i ] with leading term( T − s m X ) φ n = T − s m T − u (cid:48)− ( F i ) + · · · because the only degree, which is larger then u (cid:48) ( α i ) and becomes smaller afterreflection, is s m u (cid:48) ( α i ), which is not in u (cid:48) ( α i ) + Z α n .We now apply a φ m character-shift to both sides of the formula, giving( T − s m X ) φ m ,φ n = (cid:0) T − s m T − u (cid:48)− ( F i ) (cid:1) φ m + · · · By Proposition 5.19 we know the right-hand side shift explicitly, so( T − s m X ) φ m ,φ n = ( q m − q − m ) φ m ( F m ) · T − u (cid:48)− ( F i ) K − m + · · · which concludes the proof. (cid:3) We remark that without multiplicity zero we cannot control on which expressionwe have to apply the inductive assumption in order to get precisely the root vectorin the last line. For example, the character-shift could be zero or (since we apply T − m ) 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 induc-tive assumptions of the proof, compare Example 5.15: The new PBW generator in U − [ s n u (cid:48) s i ] is T − s n T − u (cid:48)− ( F i ), which is in degree s n u (cid:48) ( α i ) = β , while the new degree in the graded algebra is u (cid:48) ( α i ) = β + α n . This is only possible, because (apparently, byinduction) some element Y in degree β + α n = u (cid:48) ( α i ) = s n u (cid:48) ( α l ) is already presentin U − [ s n u (cid:48) ], with character-shift Y φ n = Y + T − s n T − u (cid:48)− ( F i ) + · · · . Together with T − s n T − u (cid:48)− ( F i ) this makes possible the linear combination X = Y − T − s n T − u (cid:48)− ( F i ) inmixed degrees, where the leading terms in the character-shifts cancel and the nextleading term is in degree u (cid:48) ( α i ) = β + α n .5.5. The proof of Conjecture A for type A n , B n , C n , D n . The criteria aboveimplies that Conjecture A holds for type A n , B n , C n , D n , after some by-hand ar-guments, as we shall now discuss. Type G holds trivially by rank with Criterion1. For type E , E , E , F there are several exceptional cases, which would have tobe treated by hand. Altogether a more systematic method of proof would be muchpreferred.Assume first that Criterion 1 (Lemma 5.16) applies with case A : So in Φ wehave the simple roots of the support S = { α n , α m } and a positive root β such that˜ S = { α n , β, α m } has inner product as in A , and β i := u ( α i ) , u (cid:48) ( α j ) , β j := u ( α j )are contained in the set of positive roots generated by them. We observe β (cid:54)∈ Φ( w ),otherwise all positive roots generated by ˜ S would be already in Φ( w ). In particular β i (cid:54) = β , so we assume without loss of generality β i − α m ∈ Φ. On the other hand weobserve now that β i (cid:54) = α n + β + α m (the highest root in A ), because otherwise wehave α m + β j + α n = β i or w − ( α m )+ α j + w − ( α n ) = − α i , which is absurd, because w − ( α m ) , w − ( α n ) < S ⊂ Φ( w ). It only remains β i = β + α m .Overall we have reached the situation β + α m + α n − γ − α i β i β j w − (cid:55)−→ − α i α j α m β α n − γ − α i − α j γ + α j − γ The roots β and γ have the property, that β + α m , β + α n ∈ Φ and γ + α i , β + α j ∈ Φ. There is a canonical choice for such a root in any connected root system
Definition 5.21.
For simple roots α i , α j in a connected Dynkin diagram withoutloops, there is a unique path from α i to α j , and summing along the path withoutendpoints defines a root ρ ij := ( α i − + · · · + α j − ) with the property that ρ ij + α i , ρ ij + α j ∈ Φ . Moreover it contains α i , α j withmultiplicity zero. However: If γ = ρ ij , then α n = w ( − γ ) > w ( α k ) > α k (cid:54) = α i . Thus, Criterion 2 (Lemma 5.18) implies Conjecture A, unless γ contains α i with multiplicity ≥ A n , then Criterion 1 (Lemma 5.16) returns only the case A . Sincein A n all multiplicities are ≤
1, the root γ with γ + α i ∈ Φ cannot contain α i withmultiplicity ≥
1, so Criterion 2 applies as discussed above. Alternatively we myapply Criterion 3 (Lemma 5.20), because β with β i = β + α m ∈ Φ cannot contain α m with multiplicity ≥
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 thecase A and B or C or D , which are the possible subdiagrams of the Dynkindiagram. We first assume again case A and list the possible β explicitly: Fact 5.23.
By directly inspecting the root systems B n , C n , D n , we find that forgiven α n , α m the only positive roots β with β + α n , β n + α m ∈ Φ are β = ρ mn β = α m − + · · · + α n + 2 α n − + · · · β = α n − + · · · + α m + 2 α m − + · · · where α is the unique short simple root for B n , and α is the unique long simpleroot for C n , and α , α are the two short legs for D n , and where the final summandsdepend on the type of the root system. In the first and second case again Criterion 1 applies, because β contains α m with multiplicity zero. In the third case we consider σ := w ( ρ ij ), which is a positiveroot (otherwise again Criterion 2 applies, since ρ ij does not contain α i ), and whichhas the property σ − β i , σ + β j ∈ Φ. From the explicit form of β this only leaves σ = α m , so γ = ρ ij . But then again Criterion 2 applies as for A n .Assume now type B n or C n with case B or C . So in Φ we have the short andlong simple roots of the support S = { α n , α m } and a positive (long or short) root β such that ˜ S = { α n , β, α m } has inner product as in B or C . We remark that bydirect inspection of the root system this fixes β to be simply the sum of all simpleroots between α n , α m , and in particular both α n , α m appear in β with multiplicityzero, 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 . So in Φ we have the simple roots of thesupport S = { α n , α m , α t } and a positive root β such that ˜ S = { α n , α m , α t , β } hasinner product as in D . Again direct inspection of the root system shows that then β is a sum of adjacent simple roots, and in particular contains α n , α m , α t withmultiplicity zero. So Criterion 3 implies Conjecture A. Corollary 5.25.
Conjecture A holds for type D n . The graded algebra determining the representation theory
Conjecture B.
Our ultimate hope is, that the description of the gradedalgebra in Conjecture A and Corollary 5.4, which we have proven in many casesdetermines precisely which combinations w + , w − give triangular Borel subalgebras: Conjecture B.
An arbitrary triangular right coideal subalgebra C , i.e. of the form U − [ w − ] φ − k [ L ] S ( U + [ w + ]) φ + , is a Borel subalgebra iff w + w (cid:48)− − = w with (cid:96) ( w + ) + (cid:96) ( w (cid:48)− − ) = (cid:96) ( w ) . Example 6.1.
In the homogeneous case φ + = φ − = 0 where w (cid:48)− = w − the condi-tion is clear: If (cid:96) ( w + ) + (cid:96) ( w (cid:48)− − ) > (cid:96) ( w + w (cid:48)− − ) then there are common roots α, − α in Φ + ( w + ) , Φ − ( w − ) , producing a full quantum subgroups U q ( sl ) , which is surely notbasic. If (cid:96) ( w + w (cid:48)− − ) < (cid:96) ( w ) 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, wesuggest that the full conjecture should first be attempted with the additional as-sumptions supp( φ + ) = supp( φ − ) and L = supp( φ + ) ⊥ .On the other hand we have so far no example of a non-triangular basic rightcoideal subalgebra, not contained in a triangular basic right coideal subalgebra. The conjecture has essentially two parts, which are both representation-theoretic:Proving that (cid:96) ( w (cid:48)− − w + ) < (cid:96) ( w (cid:48)− ) + (cid:96) ( w + ) implies non-basic and (cid:96) ( w (cid:48)− − w + ) = (cid:96) ( w (cid:48)− ) + (cid:96) ( w + ) implies basic. From these two it would then follow that ConjectureB characterizes basic right coideal subalgebras that are at least maximal amongall triangular basic right coideal subalgebras. We now prove the first of the firststatement in general using the Conjecture A, where proven.The second part we can so far only prove in special cases like A , A below orsupp( φ + ) = supp( φ − ) = Φ + ( w + ) ∩ Φ( w − ) by explicit knowledge of the algebra. Itwould 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 Bin one direction, by explicitly constructing higher-dimensional representations ascomposition factors of restricted U q ( g )-representations. For example, this resultallows 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 − ∈ W let w (cid:48)− be as in the proven Conjecture A. Let w + ∈ W , such that C = U − [ w − ] φ − k [ L ] S ( U + [ w + ]) φ + with L ⊂ (supp( φ + ) ∩ supp( φ − )) ⊥ is a triangular right coideal subalgebra. If (cid:96) ( w (cid:48)− − w + ) < (cid:96) ( w (cid:48)− ) + (cid:96) ( w + ) then C is not basic. Proof.
For an arbitrary root system (cid:96) ( w (cid:48)− − w + ) < (cid:96) ( w (cid:48)− ) + (cid:96) ( w + ) implies Φ + ( w + ) ∩ Φ + ( w (cid:48)− ) (cid:54) = ∅ , so there is a root µ ∈ Φ + ( w + ) ∩ Φ + ( w (cid:48)− ). By the proven Conjec-ture A this assumption about the graded algebra gr( C ) means that in C there areelements E, F with leading terms with respect to the Z -grading the root vectors E µ K − µ and F µ . Note that while E = E µ the element F starts usually with a higherroot and is not even necessarily a character-shifted root vector.Our general strategy to construct irreducible representations of dimension > U q ( g ) representation V = L ( λ ), which we want to restrict to C and decompose into irreducible compositionfactors, and we wish to prove not all of them are one-dimensional. The action ofthe commutator [ E, F ] ∈ C on every one-dimensional C -module is trivial, so ifthe composition series of C only contains such representations, then [ E, F ] actsnilpotently on V .But E contains only terms in degree ≤ µ and F only terms in degree ≥ µ , so thecommutator acts as a lower triangular matrix with diagonam entries [ E µ K − µ , F µ ] .So we need to find a U q ( g ) module V = L ( λ ) with non-zero eigenvaules for[ E µ K − µ , F µ ] , then this proves the existence of higher-dimensional irreducible com-position factors.Claim: For any root µ there exists a finite-dimensional U q ( g ) representation V on which the commutator [ E µ , F µ ] has a non-zero eigenvalue i.e. does not act nilpo-tently. For µ simple this is easily seen from the highest-weight vector in V = L ( λ )for any ( λ, µ ) (cid:54) = 0 (and for A n in general using minuscule weights). We generalizethis approach as follows:Let µ = w ( α i ) and choose any weight λ with ( α i , λ ) <
0. Then in the irreducible U q ( g )-module L ( λ ) the weight-spaces w ( λ ) and w ( λ − α i ) are both one-dimensionalwith basis T w ( v λ ) , T w ( F α i v λ ) since they are reflections of one-dimensional weightspaces. Since E µ = T w E α i and F µ = T − w − F α i we can evaluate the commutator[ E µ , F µ ] on v := T w v λ , but things are slightly complicated by T − w − on F :[ E µ , F µ ] v = ( T w E α i )( T − w − F α i )( T w v λ ) − ( T − w − F α i )( T w E α i )( T w v λ )= T w E α i T − w T − w − F α i T w − T w v λ − v λ is a highest-weight-vectorand ( T w E α i )( T w v λ ) = T w E α i v λ = 0. Now since the weight-spaces w ( λ ) and w ( λ − α i ) are both one-dimensional and the Lusztig-automorphisms are bijective, we havescalars a, b (cid:54) = 0 with T w − T w v λ = av λ T − w T − w − ( F α i v λ ) = b ( F α i v λ ) Since we further have E α i F α i v λ = [ E α i , F α i ] v λ = K − K − q − q − v λ = q ( α,λ ) − q − ( α,λ ) q − q − v λ it follows as claimed that v = T w v λ is a non-zero eigenvector[ E µ , F µ ] v = q ( α,λ ) − q − ( α,λ ) q − q − · a · b · v λ Thus as discussed above: Since [ E µ , F µ ] acts not nilpotently on the finite-dimensional U q ( g )-module L ( λ ), the restriction of this module to C containing as leading term E µ , F µ cannot only have one-dimensional composition-factors. Thus we found ahigher-dimensional irreducible composition factor and C is not basic. (cid:3) Example A We have already determined all Borel subalgebras of U q ( sl ) in Section 3.2. Weconclude our article by treating the next case U q ( sl ) explicitly. In particular wegive a completed classification of all Borel subalgebras in this case by hand, so thatour conjectures and their impact can be checked against a more realistic example.Any triangular right coideal subalgebra is of the form C = U − [ w − ] φ − k [ L ] S ( U + [ w + ]) φ + , with L ⊥ supp( φ + ) ∩ supp( φ − )where the supports of the characters φ + , φ − consist of mutually orthogonal roots.We have proven in the previous section that (cid:96) ( w (cid:48)− − w + ) < (cid:96) ( w (cid:48) ) + (cid:96) ( w + ) impliesnot basic and conjectured that the triangular Borel subalgebras are precisely thosewith w (cid:48)− − w + = w , we expect moreover supp( φ + ) = supp( φ − ) and by maximality L = (supp( φ + ) ∩ supp( φ − )) ⊥ .These expectations would lead to the following cases: • supp( φ + ) = supp( φ − ) = {} i.e. φ + , φ i trivial. These are the homogeneousBorel algebras and we have already discussed them in Section 3.3. Theyare reflections of U − , explicitly U − [ w − ] U S ( U + [ w + ]) , w − − w + = w • supp( φ + ) = supp( φ − ) = { α } , then in particular we must have α ∈ Φ + [ w ± ] leaving the three cases w ± = s , s s , s s s with w (cid:48)± = s w ± =1 , s , s s . The relation w (cid:48)− − w + = w leaves the following three cases: w − w (cid:48)− w + Φ + ( w − ) Φ + ( w (cid:48)− ) Φ + ( w + ) s s s s { α } {} { α , α , α } s s s s s { α , α } { α } { α , α } s s s s s s { α , α , α } { α , α } { α } • supp( φ + ) = supp( φ − ) = { α } leaves only cases that are isomorphic to theformer ones by diagram automorphism α ↔ α . • supp( φ + ) = supp( φ − ) = { α } leaves the three cases w ± = s s , s s , s s s with w (cid:48)± = s w ± = s , s ,
1. The relation w (cid:48)− − w + = w leaves up todiagram automorphism the case w − = s s , w (cid:48)− = s , w + = s s . It is aagain a reflection of the former solutions. † We shall discuss these two non-homogeneous examples of right coideal subal-gebras and prove that they are basic; in the last subsection we give a by-handclassification of all basic right coideal subalgebras of U q ( sl ), using similar meth-ods as in Theorem 3.8 for U q ( sl ), which shows that these are Borel subalgebrasand that they are all of them.7.1. The Borel subalgebra U − [ s ] φ − k [( K K ) ± ] S ( U + [ s s s ]) φ + . This rightcoideal subalgebra C is generated as an algebra by¯ F = F + λ (cid:48) K − , ( K K ) ± , ¯ E = E K − + λK − E K − where φ + ( E α K − α ) = λ and φ − ( F α ) = λ (cid:48) and 0 else, such that λ + λ − = q (1 − q )( q − q − ) . Figure 1.
Picture of Φ + ( w ± ) with gray lines indicating character-shifts. The contained Weyl algebra is clearly visibleIn the Z -grading these elements have degree 0 , , α , α and the graded algebra isin accordance with the proven Conjecture A gr( C ) = k [( K K ) ± , K − ] S ( U + [ s s ].The right coideal subalgebra C contains a Weyl algebra generated by ¯ F , ¯ E from type A ⊂ A and this injection splits by the algebra surjection sending E K − (cid:55)→ X µ ∈ † The reader be advised, that reflections of right coideal subalgebras are a-priori not alwaysright coideal subalgebras. S ( U + [ s s s ]) φ + in degree µ holds[ ¯ F , X µ ] q ( α ,µ ) = 0Thus for any finite-dimensional representation V we can consider the subspacewhere E K − acts by zero and this is again a C -representation. Hence for irre-ducible V this element E K − acts by zero and the action of our algebra factorizesthrough the Weyl algebra in Example 3.6.This algebra is the smallest example of a large family of basic triangular rightcoideal subalgebras that we constructed in [LV17] for type A n . They are all aproduct of commuting copies of the quantum Weyl algebra, for every simple rootin the intersection Φ( w + ) ∩ Φ( w − ), and then filled up as much as possible with thestandard Borel subalgebra.7.2. The Borel subalgebra U − [ s s ] φ − k [( K K ) ± ] S ( U + [ s s ]) φ + . This morecomplicated right coideal subalgebra C is generated as an algebra by¯ E := E K − + λK − ¯ F := F + λ (cid:48) K − K ± := ( K K ) ± ¯ E := E ( K K ) − + (1 − q − ) λE ( K K ) − ¯ F := F + ( q − − q ) λ (cid:48) F K − Figure 2.
Picture of Φ + ( w ± ) with gray lines indicating character-shifts. The Weyl algebras, one extending another, are clearly visibleThe respective graded algebra is in accordance with the proven Conjecture Agr( C ) = U − [ s ] k [( K K ) ± , K − ] S ( U + [ s s ])Here again the characters are given by φ + ( E α K − α ) = λ and φ − ( F α ) = λ (cid:48) and 0else, such that λλ (cid:48) = q (1 − q )( q − q − ) . The product of the appearing constants c := (1 − q − ) λ and c := ( q − − q ) λ (cid:48) is c c = (1 − q − ) λ ( q − − q ) λ (cid:48) = (1 − q − )( q − − q ) q (1 − q )( q − q − ) = 1We calculate the commutator relation[ E K − + c E K − , F + c F K − ] q = ( F + λ (cid:48) K − )( E K − + λK − )( q − q ) + q q − q − C are[ K, ¯ E ] = [ K, ¯ F ] = [ K, ¯ E ] = [ K, ¯ F ] = 0(1) [ ¯ E , ¯ E ] q = [ ¯ E , ¯ F ] q = 0(2) [ ¯ F , ¯ E ] q − = [ ¯ F , ¯ F ] q − = 0(3) [ ¯ E , ¯ F ] q = q q − q − E , ¯ F ] q = ¯ F ¯ E ( q − q ) + q q − q − C is basic: The algebra C contains a subalgebra (cid:104) ¯ E , ¯ F (cid:105) iso-morphic to the Weyl algebra, which acts by Example 3.6 on every irreduciblefinite-dimensional representation V e,f via its quotient algebra ¯ E ¯ F = ¯ F ¯ E = q ( q − q − )(1 − q ) , and these are all one-dimensional. But the relation reduces the com-mutator relation 5 to [ ¯ E , ¯ F ] q = 0 on any V e,f (6)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 ¯ E , ¯ F map V e,f to some subrepresentation V qe,q − f resp. V q − e,qf . Since ef (cid:54) = 0 and q is not a root of unity, finite dimension proves that ¯ E , ¯ F act nilpotently. Let V e (cid:48) ,f (cid:48) be such that ¯ E acts by zero, then by the additional relation 6 shows that¯ F preserves V e (cid:48) ,f (cid:48) 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 followsthat the basic right coideal subalgebras introduced above are maximal among all triangular basic right coideal subalgebras. With similar techniques as for U q ( sl )in Theorem 3.8 one can show: Theorem 7.1 ([Vocke16] Chapter 10) . The triangular basic right coideal subalge-bras above k [ K ± α , K ± α ] S ( U + [ w ]) U − [ s ] φ − k [( K K ) ± ] S ( U + [ w ]) φ + U − [ s s ] φ − k [( K K ) ± ] S ( U + [ s s ]) φ + are Borel subalgebras and these are all Borel subalgebras of U q ( sl ) up to reflectionand diagram automorphism. The proof idea for both assertions is that from [Vocke16] Theorem 4.11 andLemma 3.5 we have for any right coideal subalgebra in U q ( g ) a generating systemof elements, which have a unique leading term that is a root vector. Then onehas to study combinations of arbitrary elements and test them on (minuscule)representations of U q ( sl ) to rule basic out, or to show that the combination iscontained in a triangular right coideal subalgebra, because we know our examplesare the maximal ones among them. These results beyond the results on ConjectureA and B are very ad-hoc.7.4. Induction of one-dimensional characters.
We want to finally calculatethe induced representations in the two nontrivial Borel algebras above in U q ( sl )as in Section 4.Let C = U − [ s ] φ − k [( K K ) ± ] S ( U + [ s s s ]) φ + . We have seen that each irre-ducible representation is of the form V e ,f ,k , which is one-dimensional with ac-tion given by scalars e , f , k ∈ k × with e f = q ( q − q − )(1 − q ) F + λ (cid:48) K − (cid:55)→ f ( K K ) ± (cid:55)→ k E K − + λK − (cid:55)→ e E K − (cid:55)→ C has a PBW-basis, we can calculate the induced module to be U q ( sl ) ⊗ C V e ,f ,k = (cid:104) F i F j K k K (cid:15) , i, j ∈ N , k ∈ Z , (cid:15) = 0 , (cid:105) k = (cid:104) F i F j K k , i, j ∈ N , k ∈ Z (cid:105) k This is (up to K (cid:15) , which depends on the chosen lattices) the right coideal sub-algebra ˜ C = U − [ s s ] k [ K ], acting on itself by left-multiplication, and this ac-tion is extended to U q ( sl ) acting on ˜ C . A similar result follows easily wheneverΦ + ( w − ) ∩ Φ + ( w + ) = supp as we show in [LV17]. Let C = U − [ s s ] φ − k [( K K ) ± ] S ( U + [ s s ]) φ + . We have seen that each irre-ducible representation is of the form V e ,f ,k , which is one-dimensional with ac-tion given by scalars e , f , k ∈ k × with e f = q ( q − q − )(1 − q ) E K − + λK − (cid:55)→ e F + λ (cid:48) K − (cid:55)→ f ( K K ) ± (cid:55)→ k E ( K K ) − + (1 − q − ) λE ( K K ) − (cid:55)→ F + ( q − − q ) λ (cid:48) F K − (cid:55)→ C has a PBW-basis, we can calculate the induced module to be U q ( sl ) ⊗ C V e ,f ,k = (cid:104) F i E j K k K (cid:15) , i, j ∈ N , k ∈ Z , (cid:15) = 0 , (cid:105) k = (cid:104) F i E j K k , i, j ∈ N , k ∈ Z (cid:105) k This is (up to K (cid:15) , which depends on the chosen lattices) the Hopf subalgebra˜ C = U q ( sl ), acting on itself by left-multiplication, and this action is extended to U q ( sl ) acting on ˜ C .In both cases our results in Section 4 state that any finite-dimensional U q ( sl )representation appears as quotient of the induced representation for some specificvalues ( e , f , k ). The tool in Lemma 4.9 shows again that for generic values theinduced representations are irreducible.8. Outlook
Our efforts are far from being concluded. We wish to point out from our per-spective difficult points that need to be resolved in future research:(1) Conjecture A should be proven for all g . That is, α i = α j in the inductionstep for any s i w . As we have proven in Sec. 5.5, this holds by a combinationof three criteria from root system combinatorics to explicit calculations inmost cases by uniques, except a few exceptions in E , E , E with highmultiplicity. These could in principal be treated by hand, by computingthe character-shifts in non-multiplicity-zero cases of Criterion 3. However,it would be much more satisfying to have a systematic proof.(2) Conjecture B should be proven in the open direction, and the graded alge-bra gr( C ) together with Conjecture A should be used more systematically.(3) Similarly, we would expect that the graded algebra gr( C ) gives much infor-mation about the structure of the induced module. In essence, one wouldwant to use the preimage of gr( C ) much like the Cartan part for a usualVerma module. On should also try to prove standard facts, for examplethat the tops are irreducible modules. (4) The explicit family of triangular right coideal subalgebras in U q ( sl n +1 ) (con-jecturally Borel) in [LV17] consisting of a positive part and several quantumWeyl algebras. It seems to be a rather uniform construction, also in higherdepth and for arbitrary g , which should be studied. All other exampleswe know (e.g. the last example in sl ) contain extensions of Weyl algebrasby Weyl algebras, and maybe they can be inductively treated. Are therecompletely different cases?(5) Beyond triangular Borel algebras, we have observed in small examples (butdo not dare to conjecture) that for a possibly non-triangular basic rightcoideal subalgebras, there always seems to be a larger triangular rightcoideal subalgebras, which is still basic. This would imply all Borel al-gebras are triangular. If not, than a counter-example would be extremelyinteresting. Up to now, the only access we have is to use the generatortheorem in [Vocke16] for arbitrary coideal subalgebras and then combinepossible generators by hand.Moreover, the following question are from our perspective interesting with respectto applications:(1) We would find it very interesting to classify Borel subalgebras for smallquantum groups when q is a root of unity.(2) For any Borel subalgebra B , do the induced modules produce a good analogof the tensor category O B ?(3) The fact that we use coideal subalgebra means that the category of rep-resentations of C is a module category over Rep( U q ( g )). What additionalstructure is the implication of this fact? One might expect some weak linkbetween the category O B and usual category O , maybe an invertible bi-module category as in [LP17]?(4) The induced modules appear for sl in [Schm96, Tesch01] in the contextof non-compact quantum group. For g , are there corresponding families ofmodules with these additional analytic data? (for example, for sl replacing C [ K, K − ] by functions on the torus).On the other hand, is our construction related to the non-standard Borelsubalgebras of affine Lie algebras [Fut94, Cox94] via Kazhdan-Lusztig cor-respondence? References [AHS10] N. Andruskiewitsch, I. Heckenberger, H.-J. Schneider:
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