Hopf actions of some quantum groups on path algebras
aa r X i v : . [ m a t h . QA ] O c t HOPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS
RYAN KINSER AND AMREI OSWALD
Abstract.
Our first collection of results parametrize (filtered) actions of a quantum Borel U q ( b ) ⊂ U q ( sl ) on the path algebra of an arbitrary (finite) quiver. When q is a root of unity, we give nec-essary and sufficient conditions for these actions to factor through corresponding finite-dimensionalquotients, generalized Taft algebras T ( r, n ) and small quantum groups u q ( sl ).In the second part of the paper, we shift to the language of tensor categories. Here we consider aquiver path algebra equipped with an action of a Hopf algebra H to be a tensor algebra in the tensorcategory of representations H . Such a tensor algebra is generated by an algebra and bimodule in thistensor category. Our second collection of results describe the corresponding bimodule categoriesvia an equivalence with categories of representations of certain explicitly described quivers withrelations. Contents
1. Introduction 12. Background and preliminaries 33. Parametrizing actions by linear algebraic data 64. Tensor categorical viewpoint 13References 221.
Introduction
Context and motivation.
Throughout the paper we work over a field k which is oftenomitted from the notation. The only assumptions on the field are that it must contain the necessaryroots of unity whenever they are referenced.The classical mathematical notion of “symmetry” can be formalized by group actions. A groupcan act directly on an object itself, or indirectly on various spaces of functions on an object. Anadvantage of the latter approach is that spaces of k -valued functions on any object (the functionsmay be restricted to satisfy some reasonable property) are k -vector spaces, allowing for the in-troduction of tools from linear algebra and representation theory in the study of symmetry. Thisnaturally leads from groups to the study of group algebras and their actions on other k -algebras.More generally, actions of Hopf algebras on k -algebras are one way to formalize the mathematicalnotion of quantum symmetry . One may see the following (highly incomplete) list of recent works asentry points to the extensive literature on the topic [CEW16, CKWZ16, EW16, EW17, CKWZ18,CKWZ19, Cli19, BHZ19, Neg19, CG20, EKW, DNN20, LNY20, CY20], and the survey articles[Kir16, Wal19]. In this paper we study actions of several families of Hopf algebras:(1.1) U q ( b ) , U q ( sl ) , T ( n, r ) , u q ( sl ) . on path algebras of quivers. Here, the first two are rank 1 quantum groups (deformed envelopingalgebras), and the latter two are certain finite-dimensional quotients of these; see Section 2.2 forprecise definitions. Our work in this paper significantly extends the main results of [KW16] (and a Mathematics Subject Classification.
Primary 16T05; Secondary 16G20, 18M99.
Key words and phrases.
Hopf action, Taft algebra, quantum group, path algebra, quiver, tensor category. subsequent partial generalization in one chapter of [Ber18]), while simultaneously connecting withthe tensor category framework for studying Hopf actions on quiver path algebras introduced in[EKW].1.2.
Summary of main results.
Below, H always refers to one of the algebras in (1.1). Our mainresults describe all filtered actions (see Assumption 2.25) of such H on path algebras of arbitraryquivers with finitely many vertices and arrows, in two different ways.In Section 3, we parametrize these actions by basic linear algebraic data. We paraphrase Theo-rems 3.13 and 3.36 as follows, where the “certain conditions” mentioned below are explicitly listedin these theorems. Theorem.
The following data determines a (filtered) Hopf action of U q ( b ) (resp., U q ( sl ) ) on apath algebra k Q , and all such actions are of this form:(i) a Hopf action of U q ( b ) (resp., U q ( sl ) ) on k Q , as described in Proposition 3.1 (resp. 3.28);(ii) a representation of G on k Q which is compatible with its k Q -bimodule structure;(iii) a linear endomorphism (resp. pair of linear endomorphisms) of k Q ⊕ k Q satisfying certainconditions. The linear endomorphism(s) mentioned in (iii) are derived from the actions of the standardskew-primitive generators of U q ( b ) and U q ( sl ), but are in some sense independent of Hopf actionon k Q in (i) . Corollaries 4.38 and 3.44 give easily verifiable criteria for an action above to factorthrough the finite dimensional quotients T ( r, n ) and u q ( sl ).In Section 4, we shift perspective following recent joint work of the first author with Etingof andWalton [EKW], viewing a path algebra with an action of H as particular case of a tensor algebrain the tensor category rep ( H ). This leads to an analysis of bimodule categories over commutativealgebras in rep ( H ) for each Hopf algebra H in (1.1), leveraging our work in Section 3. Our mainresults here give describe the structure of these bimodule categories. We provide a brief outlinebelow; see Theorems 4.26 and 4.46 and their corollaries for the precise statements. Theorem.
Consider the tensor category C = rep ( H ) and a pair of indecomposable algebras S, S ′ in C such that S = k m and S ′ = k m ′ as k -algebras. Then the bimodule category bimod C ( S, S ′ ) isequivalent to the following category of representations in each case:(a) for H = U q ( b ) , representations of Γ( q ℓ , d ) which are nilpotent on loops (see Notation 4.6)(b) for H = T ( r, n ) , representations of Γ T (see Notation 4.34)(c) for H = U q ( sl ) , representations of Γ ′ ( q ℓ , q , d ) which are nilpotent on loops (see Notation4.43)(d) for H = u q ( sl ) , representations of Γ ′ T (see Notation 4.51).Each algebra above is presented as a quotient of a path algebra (via explicitly given relations) of aquiver whose connected components are of the form shown in (4.2) , (4.3) , (4.42) , and (4.52) . One immediate consequence of these equivalences is that the bimodule category in each caseis independent of the specific H -actions on S, S ′ , since the corresponding algebras are (Corollary4.30). Another consequence is that classifying H -actions even on very small quivers will usuallybe “at least as complicated as” classification of representations of the free associative k -algebra k h x, y i , or equivalently, pairs of matrices up to simultaneous conjugation. This is because thealgebras appearing in our theorems of Section 4 typically have finite-dimensional algebras of wildrepresentation type as quotients. It seems to us that classification up to Morita equivalence in rep ( H ) should also be difficult, but we do not see how to make that precise. A notable exceptionis when H = T ( n, r ) is a generalized Taft algebra, in which case our results show that the relevantbimodule categories can actually be of finite representation type, as they can be equivalent torepresentation categories of Nakayama algebras. OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 3
Acknowledgements.
The authors thank Chelsea Walton for valuable feedback on the first draftof this manuscript. 2.
Background and preliminaries
Hopf algebras and their actions.
We refer the reader to standard texts such as [Mon93,DNR01, Rad12] for background on Hopf algebras and Hopf actions, and we use standard notationsuch as ∆ for the comultiplication and ε for the counit. The term algebra in this paper alwaysrefers to an associative k -algebra with identity.Recall that given a Hopf algebra H and an algebra A , a (left Hopf ) action of H on A consistsof a left H -module structure on A satisfying:(a) h · ( pq ) = P i ( h i, · p )( h i, · q ) for all h ∈ H and p, q ∈ A , where ∆( h ) = P i h i, ⊗ h i, , and(b) h · A = ε ( h )1 A for all h ∈ H .In this case we also say that A is a left H -module algebra . This is equivalent to A being an algebrain the tensor category rep ( H ), the category of finite-dimensional representations of H .2.2. Generalized Taft algebras and related quantum groups.
In this section we define thespecific Hopf algebras whose actions are studied in this paper. We omit mention of the counitssince the conditions they impose throughout the paper are always trivial to check. The symbol q always denotes an element of k , with further restrictions depending on the algebra. Definition 2.1.
Let q ∈ k with q = 0 , ±
1. Following [Maj02, Ex. 1.5], the Hopf algebra U q ( b ) has k -algebra generators x, g, g − and relations(2.2) gg − = 1 = g − g, gxg − = qx. The comultiplication is given by(2.3) ∆( g ) = g ⊗ g, ∆( x ) = 1 ⊗ x + x ⊗ g. If q is a primitive r th root of unity and n a positive integer multiple of r , then the generalizedTaft algebra [Rad75] is the Hopf quotient T ( r, n ) := U q ( b ) / h g n − , x r i (see also [Cib93]). We write T ( n ) := T ( n, n ) for short; these are the classical Taft algebras [Taf71]. (cid:3)
The Hopf algebra U q ( b ) can be viewed as a deformation of the universal enveloping algebra of b . We now define a deformation of the universal enveloping algebra of sl , and a certain finite-dimensional Hopf quotient. Definition 2.4.
Following [Maj02, § U q ( sl ) is the k -algebra with generators E, F, K, K − subject to relations(2.5) KEK − = q E, KF K − = q − F, [ E, F ] = K − K − q − q − along with KK − = 1 = K − K . The comultiplication is given by(2.6) ∆( E ) = 1 ⊗ E + E ⊗ K, ∆( F ) = K − ⊗ F + F ⊗ , ∆( K ) = K ⊗ K. When q is a primitive n th root of unity with n > n odd, we also have the small quantumgroup u q ( sl ), or Frobenius-Lusztig kernel , as the Hopf quotient U q ( sl ) / h K n − , E n , F n i [Maj02,Ex. 6.4]. (cid:3) The algebras U q ( b ) and T ( n ) are analogues of Borel subalgebras in U q ( sl ) and u q ( sl ), respec-tively. Namely, we have isomorphisms of Hopf algebras h E, K i ≃ U q ( b ) where K ↔ g, E ↔ x h F, K i ≃ U q − ( b ) where K ↔ g, F ↔ g − x, (2.7)and similarly u q ( sl ) has two subalgebras isomorphic to Taft algebras. RYAN KINSER AND AMREI OSWALD
Skew-primitives and q -integers. Following [Maj02, § q = 0 , ± < m < n as(2.8) [ m ] q = 1 − q m − q , (cid:20) nm (cid:21) q = [ n ] q ![ m ] q ![ n − m ] q ! , where [ k ] q ! = [ k ] q [ k − q · · · [2] q [1] q for k ∈ Z > . We observe that for 0 < m < n we have(2.9) q n = 1 = ⇒ (cid:20) nm (cid:21) q = 0 . By convention we interpret (cid:2) nm (cid:3) q = 1 when m = 0 or n .Let k, l be positive integers. The set of weak compositions of k of length l is(2.10) W C ( k, l ) = ( λ = ( λ , λ , . . . , λ l ) ∈ Z l ≥ (cid:12)(cid:12)(cid:12)(cid:12) l X i =1 λ i = k ) , and for λ ∈ W C ( k, l ), we write | λ | = k . We denote by e i ∈ W C (1 , l ) the standard basis vectorwhich is 1 in coordinate i and 0 elsewhere. For λ ∈ W C ( k, l ), the q -multinomial coefficient isdefined as(2.11) (cid:20) kλ (cid:21) q = [ k ] q ![ λ ] q ! · · · [ λ l ] q ! . Partial sums of the entries of λ ∈ W C ( k, l ) frequently appear in our formulas, so we define(2.12) λ i = λ + λ + · · · + λ i − . (and take λ = 0). These satisfy the following recursion, which is a q -multinomial analogue ofPascal’s formula for binomial coefficients:(2.13) (cid:20) kλ (cid:21) q = k X i =1 q λ i (cid:20) k − λ − e i (cid:21) q for λ ∈ W C ( k, l ) . Notation 2.14.
Let
G, X ∈ H be two elements of an algebra H and λ ∈ W C ( k, l + 1). We definethe element G ⊗ λ X of H ⊗ l +1 by G ⊗ λ X := X λ ⊗ G λ X λ ⊗ G λ + λ X λ ⊗ · · · ⊗ G λ + ··· + λ l X λ l +1 = G λ X λ ⊗ G λ X λ ⊗ G λ X λ ⊗ · · · ⊗ G λ l +1 X λ l +1 . (2.15)For example, we have for each 1 ≤ i ≤ l and standard basis vector e i the element(2.16) G ⊗ e i X := 1 ⊗ · · · ⊗ ⊗ X ⊗ G ⊗ · · · ⊗ G where X appears in factor i . Lemma 2.17.
Let
G, X be elements of an algebra such that GX = qXG for some q ∈ k × , and let λ ∈ W C ( k, l + 1) . Then we have ( G ⊗ e i X )( G ⊗ λ X ) = q − λ i G ⊗ e i + λ X. Proof.
Recalling the notation in (2.15) and (2.16), we have(2.18) ( G ⊗ e i X )( G ⊗ λ X ) = G λ X λ ⊗ G λ X λ ⊗· · ·⊗ XG λ i X λ i ⊗ G λ i +1 X λ i +1 ⊗· · ·⊗ G λ l +1 X λ l +1 . Then applying XG λ i = q − λ i G λ i X in the i th tensor factor and pulling the resulting q − λ i to thefront, we arrive at(2.19) q − λ i (cid:16) G λ X λ ⊗ G λ X λ ⊗ · · · ⊗ G λ i X λ i ⊗ G λ i +1 X λ i +1 ⊗ · · · ⊗ G λ l +1 X λ l +1 (cid:17) which is equal to q − λ i G ⊗ e i + λ X by definition. (cid:3) OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 5
We prove a general combinatorial formula which can be applied in our setting. Recall that∆ l : H → H ⊗ l +1 is the map obtained by l applications of ∆, which is well defined by the coasso-ciativity axiom of a Hopf algebra. Proposition 2.20.
Let X be a (1 , G ) -skew primitive element in a Hopf algebra such that GX = qXG for some q ∈ k × . For any two integers l, k ≥ , we have in H ⊗ l +1 : (2.21) ∆ l ( X k ) = X λ ∈ W C ( k,l +1) (cid:20) kλ (cid:21) q − G ⊗ λ X Proof.
We prove the statement by induction on k (for l arbitrary). For k = 1, it follows easily fromskew-primitivity of X and the notation (2.16) that∆ l ( X ) = l +1 X i =1 G ⊗ e i X Since
W C (1 , l + 1) = { e i } l +1 i =1 and the binomial coefficients are all equal to 1, this confirms the basecase.Now assume that the formula (2.21) is true for some k ≥
1. Since ∆ is an algebra morphism, wehave by the induction hypothesis that∆ l ( X k +1 ) = ∆ l ( X )∆ l ( X k ) = l +1 X i =1 G ⊗ e i X ! X λ ∈ W C ( k,l +1) (cid:20) kλ (cid:21) q − G ⊗ λ X = l +1 X i =1 X λ ∈ W C ( k,l +1) (cid:20) kλ (cid:21) q − ( G ⊗ e i X )( G ⊗ λ X )= l +1 X i =1 X λ ∈ W C ( k,l +1) (cid:20) kλ (cid:21) q − q − λ i G ⊗ e i + λ X (2.22)where the last equality uses Lemma 2.17. For each fixed µ ∈ W C ( k + 1 , l + 1), we group togetherall pairs in the sum above ( i, λ ) such that e i + λ = µ , noting that this partitions the summandsabove. The coefficient of G ⊗ µ X is then(2.23) X ( i,λ ) e i + λ = µ (cid:20) kλ (cid:21) q − q − λ i = (cid:20) k + 1 µ (cid:21) q − where the last equality is (2.13). This completes the induction step and the proposition is proven. (cid:3) Quivers, path algebras, and their representations.
In this section, we establish notationand terminology for quivers and path algebras. More detailed introductions to the topic can befound in textbooks such as [ASS06, Sch14, DW17]. Recall that a quiver Q = ( Q , Q , s, t ) is aquadruple consisting of a finite set of vertices Q , a finite set of arrows Q , and two functions s, t : Q → Q producing the source and target of each arrow, respectively. We may visualize thearrows as follows: s ( a ) t ( a ) . a We omit the parentheses when possible to lighten the notation, writing sa and ta .The path algebra k Q is the associative k -algebra whose underlying k -vector space has as its basisthe set of all paths in Q , with the product of two paths being concatenation whenever possible RYAN KINSER AND AMREI OSWALD (read left to right), and zero otherwise. Therefore, k Q has a direct sum decomposition as a vectorspace k Q = ∞ M l =0 k Q l , where k Q l is the subspace of k Q generated by the set Q l of all paths of length l . This decompositionmakes k Q a graded k -algebra. For i ∈ Q , we write e i ∈ k Q for the idempotent corresponding tothe length 0 path at i .We note that k Q is a semisimple k -algebra and k Q a k Q -bimodule, inducing a natural iso-morphism of the path algebra k Q with the tensor algebra T k Q ( k Q ). Put another way, k Q isisomorphic to the quotient of the free associative k -algebra on generators { e i } i ∈ Q ∪ Q , by therelations:(2.24) e i e j = δ i,j e i for all i ∈ Q , e sa a = a = ae ta for all a ∈ Q . We use the following slightly unconventional definition, with motivation explained below. A representation V of a quiver Q is an assignment of a finite dimensional vector space V i to eachvertex i ∈ Q along with a linear map V a : V ta → V sa for each arrow a ∈ Q (note that themap goes in the opposite direction of the arrow). Equivalently, a representation of Q is a finitedimensional right k Q -module, or a contravariant functor from the free category on Q to the categoryof finite-dimensional vector spaces. The reason for this convention is that we want to keep the samedefinition of path algebra as in [KW16] which reads paths from left to right, while using the standardconvention for composition of linear maps (reading right to left) in a representation.Given two representations V and W of Q , a morphism of quiver representations ϕ : V → W isan assignment of a linear map ϕ ( i ) : V i → W i to each i ∈ Q so that ϕ ( sa ) ◦ V a = W a ◦ ϕ ( ta ).2.5. Hopf actions on path algebras.
We note once and for all that we restrict to actions satis-fying the following assumption throughout the paper. Let H be a Hopf algebra, Q a quiver, andset S = k Q and V = k Q , so that k Q = T S ( V ). Assumption 2.25.
We assume that Hopf actions on path algebras in this paper preserve theascending filtration by path length: (2.26) S ⊂ S ⊕ V ⊂ S ⊕ V ⊕ ( V ⊗ V ) ⊂ · · · . Under this assumption, we make the following elementary observations on parametrization ofHopf actions of H on T S ( V ). Let G := G ( H ) be the group of grouplike elements of H below. Lemma 2.27. (i) Suppose an action of H on T S ( V ) is given. Then this action is completelydetermined by the H -module structure of S ⊕ V .(ii) Conversely, any H -module structure on S ⊕ V uniquely extends to an H -module structureon the algebra T S ( V ) , which is a Hopf action exactly when it satisfies axioms (a) and (b) ofSection 2.1.(iii) Any action of G on S is induced by a permutation action of G on Q .Proof. Parts (i) and (ii) follow from the fact that
S, V generate T S ( V ) as an algebra. Part (iii)follows from the fact that a commutative semisimple algebra has a unique complete set of orthogonalidempotents, and an algebra automorphism must permute this set. (cid:3) Parametrizing actions by linear algebraic data
In this section we parametrize actions of the algebras (1.1) on path algebras of quivers, in termsif basic linear algebraic data.
OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 7
Actions of U q ( b ) and Taft algebras on products of k . We first consider quivers withoutarrows (i.e. algebras of the form k × k × · · ·× k ). The following result and its corollaries parametrizeactions of U q ( b ) on such algebras. We write G = h g i ∼ = Z , and | q | denotes the multiplicative orderof q in k × . Proposition 3.1.
Let Q be the vertex set of a quiver.(a) The following data determines a Hopf action of U q ( b ) on k Q .(i) A permutation action of G on the set Q ;(ii) A collection of scalars ( γ i ∈ k ) i ∈ Q such that (3.2) γ g · i = q − γ i ∀ i ∈ Q . The x -action is given by (3.3) x · e i = γ i e i − γ i q − e g · i for all i ∈ Q .(b) Every action of U q ( b ) on k Q is of the form above.Proof. (A) It is immediate to check that the data given in (i) and (ii) makes k Q into a U q ( b )-module. To check we have a Hopf action, g is grouplike and acts as an algebra automorphism byassumption, so it remains to check the action of x satisfies the axiom of a Hopf action. It is enoughto consider the relations defining k Q in (2.24), so we need to check that(3.4) x · ( δ i,j e i ) = e i ( x · e j ) + ( x · e i )( g · e j ) . Assume first i = j . By substitution we get γ i e i − γ i q − e g · i = e i ( γ i e i − γ i q − e g · i ) + ( γ i e i − γ i q − e g · i )( g · e i )= γ i e i − γ i q − e i e g · i + γ i e i e g · i − γ i q − e g · i . Now the condition (3.2) gives that γ i = 0 whenever g · i = i , since q = 1. This means that thesecond and third terms above are equal to 0, and so (3.4) is verified in this case. Now assume i = j .Substitution gives 0 = e i ( γ j e j − γ j q − e g · j ) + ( γ i e i − γ i q − e g · i )( g · e j )= 0 − γ j q − e i e g · j + γ i e i e g · j − . If i = g · j , the remaining terms are 0, and if i = g · j , the relation (3.2) forces the two remainingterms to cancel, so the equation is verified either way.(B) Taking an arbitrary action of U q ( b ) on k Q , we have an action of G on Q by Lemma 2.27,and it remains to show that x acts by the formula (3.3) for scalars γ i satisfying (3.2). We have firstfrom e i = e i that(3.5) x · e i = x · ( e i e i ) = e i ( x · e i ) + ( x · e i )( g · e i ) = ( x · e i )( e i + e g · i ) , which shows that x · e i = γ i e i + γ ′ i e g · i for some γ i , γ ′ i ∈ k . For i such that g · i = i , it follows directlyfrom (3.5) that x · e i = 0 and thus we may take γ i = 0 in this case. Now assume g · i = i . Then for i = j , we have(3.6) 0 = x · ( e j e i ) = e j ( x · e i ) + ( x · e j )( g · e i ) = 0 + γ ′ i e j e g · i + γ j e j e g · i + 0 . Taking j = g · i , we find γ ′ i = − γ g · i , so that x · e i = γ i e i − γ g · i e g · i . Finally, the relation gx = qxg in U q ( b ) applied to this forces(3.7) γ i e g · i − γ g · i e g · i = γ g · i qe g · i − γ g · i e g · i , and by comparing coefficients we confirm that γ i = γ g · i q , so every action comes from data as inpart (A). (cid:3) RYAN KINSER AND AMREI OSWALD
Corollary 3.8. If q is not a root of unity, then every Hopf action of U q ( b ) on k Q factors through U q ( b ) / h g n − , x i ≃ k ( Z /n Z ) for some n ∈ Z + .Proof. Since Q is finite, every G -orbit is finite, and so (3.2) implies that every γ i = 0 if q is not aroot of unity. (cid:3) The next corollary (together with Proposition 3.1) gives a generalization of [KW16, Prop. 3.5]from ordinary Taft algebras to generalized Taft algebras.
Corollary 3.9.
Suppose q is a primitive r th root of unity and consider an action of U q ( b ) on k Q .Then for all i ∈ Q , either r | G · i ) or γ i = 0 . The action factors through the generalized Taftalgebra T ( r, n ) if and only if, for all i ∈ Q , we have G · i ) | n and either G · i ) = r or γ i = 0 .Proof. It follows from (3.2) that q − G · i ) γ i = γ i for all i ∈ Q , so either γ i = 0 or r divides G · i ).We need a preliminary computation: from Proposition 2.20 we have that(3.10) x r · e i = x r · ( e i e i ) = X λ ∈ W C ( r, (cid:20) rλ (cid:21) q − ( x λ · e i )( g λ x λ · e i ) = ( x r · e i )( g r · e i ) + e i ( x r · e i ) , where the last equality uses (2.9). This shows that x r · e i is of the form αe i + βe g r · i for some α, β ∈ k . Then we can directly compute α, β from iterated application of (3.3), using property(3.2). Namely, we have x r · e i = γ ri e i + ( − r γ i · · · γ g r − · i q − r e g r · i = γ ri e i + ( − r γ ri q − · · · q − ( r − q − r e g r · i = γ ri ( e i − e g r · i ) , (3.11)where the first equality uses that r ≤ G · i ) guarantees the e g k · i are linearly independent for1 < k < r whenever γ i = 0, The last equality above uses the identity(3.12) q − · · · q − ( r − q − r = q − r ( r +1)2 = ( r odd − r even , so that the coefficient of e g r · i is − γ ri regardless of the parity of r .Now assume the action factors through T ( r, n ). Since g n = 1 in T ( r, n ), we see that G · i )divides n for all i ∈ Q . Since x r = 0, we get from (3.11) that g r · i = i whenever γ i = 0 and thus G · i ) = r since r divides G · i ).The converse is immediate, again using (3.11). (cid:3) Actions of U q ( b ) and Taft algebras on path algebras. We now extend the results of theprevious section by adding arrows to the quivers.
Theorem 3.13.
Let Q be a quiver.(a) The following data determines a Hopf action of U q ( b ) on k Q .(i) A Hopf action of U q ( b ) on k Q (see Proposition 3.1);(ii) A representation of G on k Q satisfying s ( g · a ) = g · sa and t ( g · a ) = g · ta for all a ∈ Q .(iii) A k -linear endomorphism σ : k Q ⊕ k Q → k Q ⊕ k Q satisfying( σ σ ( k Q ) = 0 ;( σ σ ( a ) = e sa σ ( a ) e g · ta for all a ∈ Q ;( σ σ ( g · a ) = q − g · σ ( a ) for all a ∈ Q .With this data, the x -action is given on a ∈ Q by (3.14) x · a = γ ta a − γ sa q − ( g · a ) + σ ( a ) . (b) Every (filtered) action of U q ( b ) on k Q is of the form above. OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 9
Proof. (A) It is straightforward to check that the given data makes k Q ⊕ k Q into a U q ( b )-module,thus inducing a U q ( b )-module structure on k Q ∼ = T k Q ( k Q ) as in Lemma 2.27. Furthermore,assumption (ii) is equivalent to the G -action on k Q in (ii) satisfying the axioms of a Hopf action.It remains to show that the proposed action of x satisfies the axioms of a Hopf action, by checkingthat it is compatible with the relations (2.24) defining k Q .We first perform a preliminary computation that will simplify the remainder of the proof: namely,given the data of (i) and (ii) , we have:(3.15) ( x · e sa )( g · a ) + a ( x · e ta ) = γ ta a − γ sa q − ( g · a ) . This follows by direct substitution of (3.3), recalling that for any i ∈ Q , we have g · i = i implies γ i = 0.It remains to check that the stated action of x preserves the relations involving an arrow in(2.24). For the first, we get x · ( e sa a ) = e sa ( x · a ) + ( x · e sa )( g · a )= e sa ( x · e sa )( g · a ) + e sa a ( x · e ta ) + e sa σ ( a ) + ( x · e sa )( g · a ) , (3.16)and we wish to show that this is equal to (3.14). Substituting in (3.15) and simplifying using ( σ e sa ( x · e sa )( g · a ).This can be calculated by substituting (3.3):(3.17) e sa ( x · e sa )( g · a ) = γ sa e sa ( g · a ) − γ sa q − e sa e g · sa ( g · a ) . If γ sa = 0, then the term clearly vanishes. If γ sa = 0, then we know sa = g · sa (= s ( g · a )), whichalso causes the term to vanish. The other relation x · a = x · ( ae ta ) is checked similarly, and thusthe given data determines a Hopf action of U q ( b ) on k Q .(B) Taking an arbitrary action of U q ( b ) on k Q , it restricts to an action on k Q by the filtrationassumption (2.25) on our Hopf actions, giving (i) . We get (ii) from Lemma 2.27, so it remains toshow the existence of a map σ as in (iii) such that the formula (3.14) holds. Define this map by σ ( k Q ) = 0, and for a ∈ k Q set(3.18) σ ( a ) := e sa ( x · a ) − a ( x · e ta )It is immediate to see that property ( σ
3) is satisfied, from the relation xg = q − gx , and we willreturn to ( σ
2) momentarily.From the relation a = e sa a we get the expression x · a = x · ( e sa a ) = e sa ( x · a ) + ( x · e sa )( g · a )= e sa ( x · a ) + ( x · e sa )( g · a ) + a ( x · e ta ) − a ( x · e ta )= ( x · e sa )( g · a ) + a ( x · e ta ) + σ ( a ) , (3.19)verifying that (3.14) holds (via (3.15)). It remains to show property ( σ σ ( a ) = e sa σ ( a )follows directly from the definition (3.18). To get σ ( a ) = σ ( a ) e ta , we need another formula for σ :this comes from the relation a = ae ta , which gives x · a = x · ( ae ta ) = a ( x · e ta ) + ( x · a )( g · e ta )= a ( x · e ta ) + ( x · a )( g · e ta ) + ( x · e sa )( g · a ) − ( x · e sa )( g · a )= ( x · e sa )( g · a ) + a ( x · e ta ) + ( x · a )( g · e ta ) − ( x · e sa )( g · a ) . (3.20)Comparing this with the last line of (3.19), we get the expression(3.21) σ ( a ) = ( x · a )( g · e ta ) − ( x · e sa )( g · a ) , which shows that σ ( a ) = σ ( a ) e g · ta . Thus property ( σ
2) holds, and the proof is completed. (cid:3)
Corollary 3.22.
Suppose q is not a root of unity. For any action of U q ( b ) on k Q , the map σ ofTheorem 3.13 is nilpotent, and so x · a = σ ( a ) for all a ∈ Q . Proof.
From Corollary 3.8, we know that x · e i = 0 (i.e. γ i = 0) for all i ∈ Q , so that x · a = σ ( a )by (3.14). To show σ is nilpotent, we can pass to an algebraic closure of k ; to simplify the notation,we just assume without loss of generality that k is algebraically closed. For any λ ∈ k , consider thegeneralized eigenspace(3.23) ( k Q ) λ = (cid:8) a ∈ k Q | ( g − λ M · a = 0 for some M ≫ (cid:9) , noting that repeated application of property ( σ
3) gives σ n (( k Q ) λ ) ⊆ ( k Q ) q n λ for n ≥
0. Consid-ering the decomposition into generalized eigenspaces, k Q = M λ ∈ k ( k Q ) λ , it enough to show that each ( k Q ) λ is annihilated by some power of σ . If λ = 0 we are done, andif λ = 0, we get ( k Q ) q n λ ∩ ( k Q ) q m λ = 0 for n = m since q is not a root of unity. Thus for fixed λ , we must have that ( k Q ) q n λ = 0 for n ≫
0, since k Q is finite dimensional. This shows that σ is nilpotent. (cid:3) We note that the case of r = n in the following corollary was first proven in the Ph.D. thesis ofAna Berrizbeitia [Ber18]. Corollary 3.24.
Suppose q is a primitive r th root of unity. In this case, an action of U q ( b ) on k Q factors through a generalized Taft algebra T ( r, n ) if and only if the following hold.(1) The action on k Q factors through T ( r, n ) .(2) For all a ∈ Q we have (3.25) γ rsa g r · a − γ rta a = σ r ( a ) . (3) The element g n acts as the identity on all of k Q .Proof. The same reasoning used in (3.10), applied to the relation a = e sa a , gives(3.26) x r · a = ( x r · e sa )( g r · a ) + e sa ( x r · a ) . Note, if the action of U q ( b ) on k Q factors through T ( r, n ), the action of k Q factors through T ( r, n ),and therefore Corollary 3.9 holds in either direction of the corollary. Thus, we can substitute for x r · e sa using (3.11) to get x r · a = γ rsa e sa ( g r · a ) − γ rsa e g r · sa ( g r · a ) + e sa ( x r · a ) = e sa ( x r · a ) , where the second equality uses that either γ sa = 0 or sa = g r · sa (= s ( g r · sa )) to get the first twoterms to vanish or cancel. A similar computation shows that x r · a = ( x r · a ) e ta .Knowing now that x r · a = e sa ( x r · a ) e ta , we can explicitly calculate it. We see that the sumresulting from r applications of the formula (3.14) results in a linear combination of terms of theform σ i ( g j · a ) with i, j ∈ Z ≥ and i + j ≤ r where the scalars contain positive powers of γ sa or γ ta for all but the σ r ( a ) term.For any term with a positive power of γ sa and j < r , we have s ( σ i ( g j · a )) = s ( g j · a ). Then, byCorollary 3.9, either γ sa = 0 or s ( g j · a ) = sa . For any terms containing only positive powers of γ ta and no positive powers of γ sa , we have that i + j < r . Since t ( σ i ( g j · a ) = t ( g i + j · a ), Corollary 3.9implies that either γ ta = 0 or t ( g i + j · a ) = ta .Since x r · a ∈ e sa k Q e ta , this reduces us to consideration of the pairs (0 , , ( r, , (0 , r ). Thecoefficients of the corresponding terms are computed using the same method as in (3.11) to get(3.27) x r · a = γ rta a − γ rsa g r · a + σ r ( a ) , which shows that the action factors through T ( r, n ) if and only if (3.25) holds. (cid:3) OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 11
Actions of U q ( sl ) and u q ( sl ) on path algebras. We again start with parametrizing actionsof U q ( sl ) and u q ( sl ) on quivers without arrows. We let G = h K i . Proposition 3.28.
Let Q be the vertex set of a quiver.(A) The following data determines a Hopf action of U q ( sl ) on k Q .(i) A permutation action of G on the set Q ;(ii) Two collection of scalars ( γ Ei ∈ k ) i ∈ Q and ( γ Fi ∈ k ) i ∈ Q such that (3.29) γ EK · i = q − γ Ei and γ FK · i = q γ Fi ∀ i ∈ Q . For each i ∈ Q such that K · i = i , these scalars must furthermore satisfy (3.30) γ Ei γ Fi = − q (1 − q ) . The action is given by E · e i = γ Ei e i − γ Ei q − e K · i for all i ∈ Q . F · e i = γ Fi e K − · i − γ Fi q e i for all i ∈ Q . (3.31) (B) Every action of U q ( sl ) on k Q is of the form above.Proof. (A) The isomorphisms (2.7) along with Proposition 3.1 show that the given data defineactions of the subalgebras h K, E i and h K, F i on k Q . These subalgebras together generate U q ( sl ),and the only compatibility condition for such a pair of actions to define an action of U q ( sl ) is thatthe rightmost relation of (2.5) holds. It can be directly computed using Proposition 3.1 that(3.32) ( EF − F E ) · e i = γ Ei γ Fi (1 − q )( e K · i − e K − · i ) , which must be equal to ( q − q − ) − ( e K · i − e K − · i ). If K · i = i then this condition is vacuous. Ifnot, then equality of these two expressions is equivalent the condition (3.30).(B) This is immediate from Proposition 3.1(B) along with the observations above. (cid:3) Remark 3.33.
We note that for actions in the above Proposition where K does not fix anyvertices, equation (3.30) implies that each of γ Ei , γ Fi is nonzero and determines the other, so theaction is completely determined by its restriction to a quantum Borel, an analogue of a result in[MS01]. It may also be interesting to compare the present work to actions of U q ( sl ) and relatedalgebras constructed in [MS90]. (cid:3) Corollary 3.34. If q is not a root of unity, then every Hopf action of U q ( sl ) on k Q factorsthrough U q ( sl ) / h K − , E, F i ≃ k ( Z / Z ) .Proof. From the isomorphisms (2.7) and Corollary 3.8, we see that γ Ei = γ Fi = 0 for all i ∈ Q , sothe action factors through U q ( sl ) / h K n − , E, F i for some n ∈ Z + . But then (3.30) can never besatisfied since the right hand side is nonzero, so K · i = i for all i ∈ Q . (cid:3) Corollary 3.35.
Let n = | q | and suppose n > and odd, and we have an action of U q ( sl ) on k Q .Then for all i ∈ Q , we have either G · i ) = 1 , , or is divisible by n . Such an action factorsthrough u q ( sl ) if and only if every G -orbit on Q has 1 or n vertices.Proof. Since n is odd, | q | = n as well. Applying Corollary 3.9 to the restricted actions of h E, K i ≃ U q ( b ) and h F, K i ≃ U q − ( b ) from (2.7), we see that for each i ∈ Q , either n divides G · i )or γ Ei = γ Fi = 0. If γ Ei = γ Fi = 0, then (3.30) can never be satisfied since the right hand side isnonzero, so we have K · i = i in this case. This proves the first statement.Now the U q ( sl ) action factors through u q ( sl ) if and only if the restricted actions of the sub-algebras h K, E i and h K, F i factor through h K, E i / h K n − , E n i and h K, F i / h K n − , F n i , whichare each isomorphic to the Taft algebra T ( n ). Corollary 3.9 tells us that this happens if and onlyif G · i ) | n and G · i ) = n when γ Ei = 0 or γ Fi = 0. Using the first part of this corollary, G · i ) | n is equivalent to G · i ) = 1 or n (since n is odd). For the second condition, if γ Ei = 0or γ Fi = 0 then G · i ) ≥ n > G · i ) = n by the first part ofthis corollary, and thus this condition is automatically satisfied. (cid:3) We now proceed to parametrize actions of U q ( sl ) and u q ( sl ) on arbitrary quivers. Theorem 3.36.
Let Q be a quiver. (A) The following data determines a Hopf action of U q ( sl ) on k Q .(i) A Hopf action of U q ( sl ) on k Q (as in Proposition 3.28);(ii) a representation of G on k Q satisfying s ( K · a ) = K · sa and t ( K · a ) = K · ta for all a ∈ Q ;(iii) a pair of k -linear endomorphisms σ E , σ F : k Q ⊕ k Q → k Q ⊕ k Q satisfying( σ σ • ( k Q ) = 0 for • ∈ { E, F } ;( σ σ • ( a ) = e sa σ • ( a ) e K · ta for • ∈ { E, F } and all a ∈ Q ;( σ σ E ( K · a ) = q − K · σ E ( a ) and σ F ( K · a ) = q K · σ F ( a ) for all a ∈ Q ;( σ γ Esa γ Fsa (1 − q ) K · a − γ Eta γ Fta (1 − q ) a + q σ E ( σ F ( a )) − σ F ( σ E ( a ))= ( q − q − ) − ( K · a − a ) for all a ∈ Q .The action is given on a ∈ Q by E · a = γ Eta a − γ Esa q − ( K · a ) + σ E ( a ) F · a = γ Fta ( K − · a ) − γ Fsa q a + K − · σ F ( a )(3.37) (B) Furthermore, every (filtered) action of U q ( sl ) on k Q with Q connected is of the form above.Proof. The isomorphisms (2.7) along with Theorem 3.13(A) show that the data given in (i), (ii),(iii) but omitting ( σ
4) define actions of the subalgebras h K, E i and h K, F i via (3.37) on k Q , andthe restrictions of these actions to h K i are the same. These subalgebras together generate U q ( sl ),and the only compatibility condition for such a pair of actions to define an action of U q ( sl ) is thatthe rightmost relation of (2.5) holds for the action. It can be directly computed using Theorem3.13 that ( EF − F E ) · a = γ Esa γ Fsa (1 − q ) K · a − γ Eta γ Fta (1 − q ) K − · a + q K − · σ E ( σ F ( a )) − K − · σ F ( σ E ( a )) , (3.38)so the given data defines an action of U q ( sl ) on k Q if and only if this is equal to(3.39) ( q − q − ) − ( K · a − K − · a )for all a ∈ Q . Acting by K on both sides of this equality gives us condition ( σ U q ( sl ) on k Q restricts to actions of the subalgebras h K, E i and h K, F i , whichare of the form stated in (A) by the isomorphisms (2.7) and Theorem 3.13(B). (cid:3) Remark 3.40. If a ∈ Q satisfies G · sa ) , G · ta ) >
2, then ( σ
4) in Theorem 3.36 simplifies to(3.41) q σ E σ F ( a ) = σ F σ E ( a ) . This is a direct result of substituting (3.30) in to ( σ
4) and simplifying. (cid:3)
Corollary 3.42.
Suppose q is not a root of unity. For any action of U q ( sl ) on k Q , both the maps σ E and σ F are nilpotent, and E · a = σ E ( a ) and F · a = K − · σ F ( a ) for all a ∈ Q . Furthermore,these maps satisfy (3.43) q σ E ( σ F ( a )) − σ F ( σ E ( a )) = K · a − aq − q − for all a ∈ Q . Proof.
This is immediate from Corollary 3.22 and Theorem 3.36. (cid:3)
Corollary 3.44. If q is a primitive n th root of unity with n odd and n > , the action of U q ( sl ) on k Q factors through u q ( sl ) if and only if the following conditions hold. OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 13 (1) K n acts as the identity on all of k Q .(2) The action of U q ( sl ) on k Q factors through u q ( sl ) (see Corollary 3.35).(3) For any a ∈ Q , we have (3.45) (( γ • sa ) n − ( γ • ta ) n ) a = ( σ • ) n ( a ) , for • ∈ { E, F } .Proof. The action of U q ( sl ) factors through u q ( sl ) if and only if the actions of the Borel subalgebras h E, K i and h F, K i factor through T ( n ). The statement follows from Corollary 3.24. (cid:3) Tensor categorical viewpoint
We now turn to the language of tensor categories, and in the case of Taft algebras, connect ourresults with work of Etingof and Ostrik on exact algebras in the category rep ( T ( n )).4.1. Hopf actions and tensor categories.
The study of Hopf actions on quiver path algebrasfalls within the more general framework of tensor algebras in tensor categories, as in [EKW]. Here,the relevant tensor categories are C = rep ( H ) where H is one of the Hopf algebras of (1.1). Thesecategories are not all finite in the sense of [EO04, § H be a Hopf algebra, Q a quiver, and let S = k Q and V = k Q , considered as an S -bimodule, so that k Q ∼ = T S ( V ). We say an H -action on T S ( V ) is graded if it preserves the pathlength grading, which is equivalent to each of S and V being H -stable. A graded H -action makes S into an algebra in C and V an S -bimodule in C . In the language of [EKW], we say T S ( V ) is a C -tensor algebra. It is furthermore a minimal, faithful C -tensor algebra if V is an indecomposable S -bimodule in C , and no two-sided ideal of S in C acts by 0 on V . Intuitively, these are the buildingsblocks of all C -tensor algebras.To study minimal, faithful tensor algebras in C we may assume S has only 1 or 2 indecomposablesummands as an algebra in C (see [EKW, Rmk 3.15]). We do not consider the notion of equivalenceof tensor algebras in this paper, so the case of S being indecomposable is subsumed by the case of S having 2 indecomposable summands, by taking S = S in the study of S - S -bimodules in C .In the following sections, we consider categories of S - S -bimodules in C , where S , S arecommutative algebras in C . We establish an equivalence between any such category of bimodules,and a subcategory of certain finite-dimensional representations of an associative k -algebra, explicitlygiven in terms of quivers with relations.4.2. The relevant quivers and bimodules.
Fix a positive integer N and µ ∈ k × . To this datawe associate an infinite quiver Q ( µ, N ) whose vertex set is k × × Z /N Z . Coming out of each vertex( λ, k ) there is a loop, and an arrow ( µλ, k + 1) → ( λ, k ). We refer to the loops as b -type arrows and the arrows ( µλ, k + 1) → ( λ, k ) as a -type arrows . When it is not clear from context, we indexthese by their target vertex. The following lemma (whose easy proof is omitted) gives the form ofthe connected components of this quiver. Lemma 4.1. If µ is not a root of unity, then the connected components of Q ( µ, N ) all have theform S ∞ , where S ∞ is the infinite quiver shown below. (4.2) S ∞ = · · · ( µλ, k + 1) b ( λ, k ) b ( µ − λ, k − b · · · a a a If µ is a root of unity, each connected component of Q ( µ, N ) has the form shown below, where S p has p := lcm( | µ | , N ) vertices. (4.3) S p = ( µ p − λ, p − b ( µ p − λ, p − b · · · ( λ, ba a aa Let Γ( µ, N ) be the quotient of k Q ( µ, N ) by the ideal generated by all relations of the form,recalling our convention from Section 2.4 of reading paths from left to right.(4.4) ba = µab. Let nlrep Γ( µ, N ) be the category of finite-dimensional representations of Γ( µ, N ) for which the lin-ear map assigned to each loop is nilpotent (but larger oriented cycles are not necessarily nilpotent).We typically denote a representation of Γ( µ, N ) by the shorthand W = ( W λ,k , A λ,k , B λ,k ), where W λ,k is the vector space associated to vertex ( λ, k ), and (recalling our contravariant convention forrepresentations from Section 2.4)(4.5) A λ,k ∈ Hom k ( W λ,k , W µλ,k +1 ) , B λ,k ∈ End k ( W λ,k )are the maps associated to the arrows a λ,k and b λ,k respectively.For a positive integer m , consider the algebra S m with action of G = h g i ≃ Z defined asfollows. We define S m = k m as a vector space, with coordinate-wise multiplication. Letting e i for 0 ≤ i ≤ m − i th standard basis vector in S m , we get a complete system of primitiveorthogonal idempotents { e , . . . , e m − } . The action of G is by g · e i = e i +1 , where we always interpretthe subscript modulo m . We identify S m with the path algebra of a quiver with m vertices and noarrows. Notation 4.6.
To study minimal and faithful rep ( U q ( b ))-tensor algebras, we fix the followingnotation. • a pair of positive integers ( m, m ′ ) • ℓ = lcm( m, m ′ ) and d = gcd( m, m ′ ) • Q := Q ( q ℓ , d ) as defined at the beginning of this subsection • Γ( q ℓ , d ) is the quotient of k Q ( q ℓ , d ) by the ideal generated by relations of the form (4.4) • N := nlrep Γ( q ℓ , d ) as defined just above (4.4) • S m , S m ′ are the algebras defined immediately above – through the end of Section 4.5, weassume these algebras have fixed U q ( b )-actions extending the G -action • B is the category of (finite-dimensional) S m - S m ′ -bimodules in rep ( U q ( b ))In the following subsections we construct mutually quasi-inverse equivalences(4.7) Φ : B → N , Ψ :
N → B . We need a little more technical notation for construction of these functors. Let R ⊂ k × be aset of coset representatives for the subgroup h q ℓ i in k × , and let ( λ, k ) be a vertex of Q . Define afunction(4.8) τ : Q → Z in two cases, depending whether q is a root of unity or not. Given ( λ, k ) ∈ Q , let λ ∈ R be thecoset representative of λ . If q is not a root of unity, there exists a unique s ∈ Z such that λ = q ℓs λ .In this case, we take τ ( λ, k ) ∈ Z minimal such that τ ( λ, k ) ≥ s and τ ( λ, k ) = k mod d . If q is aroot of unity, there exists a unique 0 ≤ τ ( λ, k ) < lcm( | q ℓ | , d ) such that both λ = q ℓτ ( λ,k ) λ and τ ( λ, k ) = k mod d . OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 15
When q is a root of unity and, we also define a “correction factor” associated to the rightmostvertex in (4.3). This is only needed for technical purposes later:(4.9) ǫ = ǫ ( q, m, m ′ , λ, j ) := ( z m if q is a root of unity and τ ( λ, k ) = lcm( | q ℓ | , d ) −
10 otherwise . Here, z , z ∈ Z are chosen so that lcm( | q ℓ | , d ) = z m + z m ′ (which is possible since the left handside is a multiple of d ).4.3. Unraveling bimodules to quiver representations.
In this section, we construct a functorΦ :
B → N which unravels a bimodule to extract a quiver representation that minimally encodesthe data defining the bimodule. We will see that this functor admits a quasi-inverse in the nextsection.Retaining the notation above, we fix a bimodule V ∈ B . We wish to construct a representation W ∈ N using the data from Theorem 3.13 that determines the action of U q ( b ) on V . Recall that V can be decomposed into arrow spaces(4.10) V = M i,j V ij , where V ij := e i V e j , where the superscripts are interpreted modulo m and the subscripts modulo m ′ . Each arrow space V ij is a representation of the subgroup h g ℓ i ≤ G . Decomposing as representations of h g ℓ i , we getgeneralized eigenspace decompositions(4.11) V ij = M λ ∈ k × V ij ( λ ) , V ij ( λ ) := n v ∈ V ij | ( g ℓ − λ M · v = 0 for M ≫ o . Recalling the definition of τ below (4.8), we then set(4.12) W λ,k := V τ ( λ,k ) ( λ ) . Appying Theorem 3.13 to the path algebra with U q ( b )-action k Q ≃ T S m ⊕ S m ′ ( V ), we get a linearmap σ : V → V satisfying relation ( σ σ ( V ij ( λ )) ⊆ V ij +1 ( q ℓ λ ) for all λ ∈ k × . In the case q is not a root of unity, one checks from the definition that τ ( q ℓ λ, k + 1) = τ ( λ, k ) + 1for any ( λ, k ) ∈ Q , thus we have a natural linear map(4.14) A λ,k := σ | W λ,k : W λ,k → W q ℓ λ,k +1 for each ( λ, k ).When q is a root of unity, however, we only have τ ( q ℓ λ, k + 1) = τ ( λ, k ) + 1 when τ ( λ, k ) =lcm( | q ℓ | , d ) − p −
1, so we only get maps of the form (4.14) over each rightward pointing a -typearrow in components of Q of the form (4.3). This is because the restriction of σ sends(4.15) W q ℓ ( p − λ,p − = V p − ( q ℓ ( p − λ ) = V p − ( q − ℓ λ ) −→ V p ( λ ) , with the target being G -equivariantly isomorphic (but not equal) to the intended target of V ( λ ) = W λ, . For this reason we introduced the “correction factor” ǫ in (4.9): since ǫ ≡ m and ǫ ≡ p mod m ′ , the action of g − ǫ gives a functorial isomorphism(4.16) V p ( λ ) ∼ −→ V ( λ ) . Thus the composition of (4.15) and (4.16) gives a linear map(4.17) A q ℓ ( p − λ, p − W q ℓ ( p − λ, p − → W λ, associated to the lower, leftward pointing arrow in a component of Q of the form (4.3). We also have a nilpotent endomorphism B λ,k = ( g ℓ − λ (cid:12)(cid:12) W λ,k of each W λ,k . With this, wedefine a representation of Q by Φ( V ) = ( W λ,k , A λ,k , B λ,k ). The relation ( σ
3) applied ℓ timesimplies that B q ℓ λ,k A λ,k = q ℓ A λ,k B λ,k for every ( λ, k ) ∈ Q , so Φ( V ) satisfies the relations (4.4), andthus Φ( V ) ∈ N . This gives the definition of Φ on objects.Let ϕ : V → V ′ be a morphism in B . Define Φ on morphisms by Φ( ϕ ) = ¯ ϕ where ¯ ϕ ( λ, k ) = ϕ | W λ,k . Proposition 4.18.
The definition above makes Φ a functor B → N .Proof.
Retaining the notation above, write Φ( V ′ ) = W ′ = ( W ′ λ,k , A ′ λ,k , B ′ λ,k ). Let σ and σ ′ be themaps which determine the action of x on V and V ′ respectively, as in Theorem 3.13.We must show that ¯ ϕ : W → W ′ is a morphism in N . Since ϕ is a morphism of bimodules,it preserves arrow spaces. Since ϕ is a morphism of U q ( b )-representations, it commutes with theaction of g ℓ and thus preserves the generalized eigenspaces of this action. It follows that ϕ | W λ,k isa map W λ,k → W ′ λ,k and that B ′ λ,k ¯ ϕ ( λ, k ) = ¯ ϕ ( λ, k ) B λ,k for every ( λ, k ) ∈ Q .For a ∈ W λ,k , we have x · ϕ ( a ) = ϕ ( x · a ), which expands via (3.14) to(4.19) γ ta ϕ ( a ) − γ sa q − g · ϕ ( a ) + σ ′ ϕ ( a ) = γ ta ϕ ( a ) − γ sa q − g · ϕ ( a ) + ϕσ ( a ) , so σ ′ ϕ ( a ) = ϕσ ( a ). Therefore, A ′ λ,k ¯ ϕ ( λ, k ) = ¯ ϕ ( q ℓ λ, k + 1) A λ,k and ¯ ϕ is a morphism in N . (cid:3) Equivalence of B and N . In this section, we construct a functor Ψ :
N → B which reversesthe process carried out by Φ, taking a quiver representation and constructing a bimodule from itin a minimal way. More precisely, we will show that Ψ is quasi-inverse to Φ.Fix W = ( W λ,k , A λ,k , B λ,k ) ∈ N . We consider each space W λ,k as a representation of h g ℓ i viathe action(4.20) g ℓ · w = λw + B λ,k ( w ) , for w ∈ W λ,k . We define vector spaces for 0 ≤ j < m ′ (recalling τ : Q → Z from the previous section) by(4.21) ˜ W j = M ( λ,k ) ∈Q τ ( λ,k )= j W λ,k . Then we define an S m - S m ′ -bimodule structure on(4.22) ˜ W = m ′ − M j =0 ˜ W j by e ˜ W e j = ˜ W j . Let H ≤ k G be the subalgebra generated by g ℓ . Since H acts on ˜ W , the space k G ⊗ H ˜ W has an induced action of k G . Moreover, we can extend the S m - S m ′ -bimodule structure on˜ W to k G ⊗ H ˜ W by e i ( g t ⊗ w ) e j = g t ⊗ ( e i − t we j − td ), which is well defined because ℓ = lcm( m, m ′ ).We now extend the action of G to an action of U q ( b ) by specifying an action of x . Let { γ i } m − i =0 and { γ ′ j } m ′ − j =0 be the scalars from Proposition 3.1 determining the actions of x on S m and S m ′ ,respectively. For w ∈ W λ,k and t ∈ Z , let j := τ ( λ, k ) and define(4.23) x · ( g t ⊗ w ) = q − t ( γ ′ j g t ⊗ w − γ q − g t +1 ⊗ w + g t + ǫ ⊗ A λ,k ( w ))where j is interpreted modulo m ′ as usual, and ǫ is the “correction factor” from (4.9).To see that this action is well-defined on k G ⊗ H ˜ W , one can check that(4.24) x · ( g ℓ ⊗ w ) − x · (1 ⊗ g ℓ · w ) = γ ′ j ( q − ℓ − g ℓ ⊗ w − γ q − ( q − ℓ − g ℓ +1 ⊗ w, where the relation q ℓ A λ,k B λ,k = B q ℓ λ,k +1 A λ,k needs to be used in the computation. The firstsummand on the right hand side always vanishes because either | q | divides m ′ , which divides ℓ , or OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 17 γ ′ j = 0 from Corollaries 3.8 and 3.9. The second summand also vanishes by the same reasoning, sothe expression (4.23) is well-defined.It follows that Ψ( W ) := k G ⊗ H ˜ W is an object of B , giving the definition of Ψ on objects. Fora morphism ϑ : W → W ′ in N , we can simply take Ψ( ϑ ) = id k G ⊗ ϑ . It is straightforward to seethen that Ψ is a functor. Proposition 4.25.
We have that Ψ is a functor N → B . Finally, we come to the main result of this section. The key idea of the proof is that the τ -functionwas carefully chosen to accomplish the unraveling, extraction, and reconstruction described at thestart of the previous two subsections. Theorem 4.26.
The functors Φ and Ψ are mutually quasi-inverse, thus the categories B and N are equivalent.Proof. For V ∈ B , we have Ψ(Φ( V )) = k G ⊗ H ˜Φ( V ). Consider the map ξ V : k G ⊗ H ˜Φ( V ) → V defined by ξ V ( g t ⊗ w ) = g t · w . We claim this is an isomorphism in B . It is straightforward to checkthat this is a morphism in B , so we need to verify it is an isomorphism of vector spaces.To see that ξ V is surjective, recall the decomposition of V from (4.10) and (4.11). It is enoughto show that each space of the form V j ( λ ) is in the image, since the action of G can then be usedto obtain that all V ij ( λ ) are in the image. If q is not a root of unity, let k ∈ Z be minimal suchthat k ≡ j mod d and k ≥ s where s is the unique integer so that λ = q ℓs λ for λ ∈ R . If q is aroot of unity, choose k to be the unique integer so that k ≡ j mod d and 0 ≤ k < lcm ( (cid:12)(cid:12) q ℓ (cid:12)(cid:12) , d ), and λ = q ℓk λ . In either case this means that τ ( λ, k ) = k , and there exist integers y , y with(4.27) τ ( λ, k ) = k = j + y m + y m ′ , and thus under the map above we have g − y m ⊗ V τ ( λ,k ) ( λ ) ∼ −→ V j ( λ ).To see that ξ V is injective, we can use its G -equivariance to reduce to considering t, k, k ′ , λ suchthat(4.28) ξ V ( g t ⊗ V τ ( λ,k ) ( λ )) = ξ V (1 ⊗ V τ ( λ,k ′ ) ( λ )) , or V tτ ( λ,k )+ t ( λ ) = V τ ( λ,k ′ ) . We want to show t ≡ ℓ and τ ( λ, k ) = τ ( λ, k ′ ). From the right hand side of (4.28) we have t ≡ m and τ ( λ, k ) + t ≡ τ ( λ, k ′ ) mod m ′ . Combining these, we find that there exist z ′ , z ′ ∈ Z such that(4.29) τ ( λ, k ) − τ ( λ, k ′ ) = z ′ m + z ′ m ′ . so τ ( λ, k ) − τ ( λ, k ′ ) is a multiple of d . When q is not a root of unity, the definition of τ impliesthat in fact τ ( λ, k ) − τ ( λ, k ′ ) = 0 and k ′ = k . When q is a root of unity, we have λ = q ℓτ ( λ,k ) λ = q ℓτ ( λ,k ′ ) λ so | q ℓ | divides τ ( λ, k ) − τ ( λ, k ′ ) as well, which shows τ ( λ, k ) − τ ( λ, k ′ ) is divisible bylcm( | q ℓ | , d ). Then the restriction that 0 ≤ τ ( λ, k ) , τ ( λ, k ′ ) < lcm( | q ℓ | , d ) forces τ ( λ, k ) = τ ( λ, k ′ )in this case as well, and thus t ≡ m ′ . This then implies that t is a multiple of ℓ , so that g t ⊗ V τ ( λ,k ) ( λ ) = 1 ⊗ V τ ( λ,k ) = 1 ⊗ V τ ( λ,k ′ ) , completing the proof that ξ V is injective. It is thenstraightforward to verify ξ V is natural in V , giving an isomorphism of functors Ψ ◦ Φ ≃ Id B .To show the other composition is the identity, we take W ∈ N , and examine the vector spaceassigned to vertex ( λ, k ) by the representation Φ(Ψ( W )) = Φ( k G ⊗ H ˜ W ). As defined in (4.12), thisis the generalized λ -eigenspace of e ( k G ⊗ H ˜ W ) e τ ( λ,k ) . We claim that this is just 1 ⊗ W λ,k ∼ = W λ,k .Indeed, if g t ⊗ W λ ′ ,k ′ were also a summand of the generalized λ -eigenspace of e ( k G ⊗ H ˜ W ) e τ ( λ,k ) ,then we would have λ ′ = λ , t = 0 mod m , and τ ( λ, k ′ ) + t = τ ( λ, k ) mod m ′ . This gives the sameequation (4.29) as above, and proceeding by the exact same argument we arrive at the conclusionthat ( λ ′ , k ′ ) = ( λ, k ) ∈ Q , and furthermore t is divisible by both m and m ′ , thus divisible by ℓ , so g t ⊗ W λ ′ ,k ′ = 1 ⊗ W λ,k . This shows that Φ(Ψ( W )) assigns the same vector space to each vertex of Q as W . Now it follows from (4.23) that the map over the arrow a λ,k in Φ(Ψ( W )) is 1 ⊗ A λ,k , and itfollows from (4.20) that the map over the arrow b λ,k in Φ(Ψ( W )) is 1 ⊗ B λ,k . This shows thatΦ(Ψ( W )) ∼ = W in N , and from there it is easy to verify the remaining details to see we have anisomorphism of functors Φ ◦ Ψ ∼ = Id N . (cid:3) Since the algebra Γ( q ℓ , d ), and thus N , only depends on | q ℓ | and d , we get the following corollary. Corollary 4.30.
The bimodule category B is (up to equivalence of categories) independent of thespecific actions of U q ( b ) on S m and S m ′ . It only depends on | q ℓ | and d , up to equivalence. Remark 4.31.
This shows that it will not be feasible to give any kind of “list” classifying inde-composable S m - S m ′ bimodules in rep ( U q ( b )), since the associated algebra Γ( q ℓ , d ) will always havefinite-dimensional quotients of wild representation type. Informally, this means that the category N will always be “at least as complicated as” the category of finite-dimensional modules over thefree associative k -algebra k h x , x i . (cid:3) Taft algebras and work of Etingof-Ostrik.
Assume here that k is algebraically closed,and q is a primitive n th root of unity. In this case, Etingof and Ostrik gave a classification of exactmodule categories over rep ( T ( n )) by classifying algebras in rep ( T ( n )) which have no nontrivial rightideals in rep ( T ( n )) [EO04, Theorem 4.10]. We begin with a dictionary between the classificationof T ( n )-actions on commutative semisimple algebras k Q in Corollary 3.9 (coming from [KW16,Prop. 3.5]), and the Etingof-Ostrik classification. This is logically independent from the rest of thepaper, included for completeness of the story. We then present our results on bimodule categoriesover generalized Taft algebras.Etingof and Ostrik found that the following list covers all algebras in the tensor category rep ( T ( n )), up to Morita equivalence as algebras in rep ( T ( n )). We only describe the algebras enoughto establish notation that uniquely identifies them, referring the reader to the proof of [EO04, The-orem 4.10] for more detail, where they proceed by considering the socle filtration of an algebra (asan object of rep ( T ( n ))).Recall that G ( T ( n )) = h g i is cyclic of order n . Below, t ranges over the positive divisors of n ,and H ≤ G denotes the unique subgroup of order t . A ( t ): denote by A ( t ) the algebra of k -valued functions on the (discrete) group G/H . This is anexact algebra in rep ( T ( n )) with x acting trivially. This commutative, semisimple k -algebrais non-semisimple and indecomposable as an algebra in rep ( T ( n )). For s ∈ G/H , we let e s be the characteristic function of the set { s } . A ( t, λ ): Taking an additional parameter λ ∈ k , we get an algebra A ( t, λ ) generated by A ( t ) and anelement y satisfying(4.32) y = X s ∈ G/H e gs ye s , y n = λ. These k -algebras are commutative if and only if t = n (studied earlier in [MS01]), and theyare all semisimple and indecomposable as algebras in rep ( T ( n )).We can now compare these to the algebras described in Proposition 3.1 and Corollary 3.9. Suchan algebra is indecomposable if and only if G acts transitively on the vertices. Up to relabeling,we have such a G -action on t vertices for each positive divisor t of n . The following translation canbe checked by direct computation. Proposition 4.33.
We have the following correspondence between indecomposable exact algebrasin rep ( T ( n )) described in Proposition 3.1 and Corollary 3.9, and those in [EO04, Theorem 4.10] : • the unique indecomposable algebra k Q in our work with | Q | = t vertices and parameters { γ i } i ∈ Q = { } is A ( n/t ) ; OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 19 • each indecomposable algebra k Q in our work with | Q | = n and parameter set { γ i } i ∈ Q is A ( n, γ − n (1 − q − ) − n ) , where γ = γ i for any choice of i ∈ Q . We now combine the results of previous sections to describe the bimodule categories over gener-alized Taft algebras in terms of quiver representations.
Notation 4.34.
We fix the following notation. • q is a primitive r th root of unity • if m = r or m ′ = r , u is the integer so that r = ud • B T is the full subcategory of B consisting of bimodules whose U q ( b ) action factors throughthe generalized Taft algebra T ( r, n ) • T is the quiver with vertices ( ζ, i ), where ζ is an ( n/ℓ ) th root of 1 and i ∈ Z /d Z , and a -typearrows ( ζ, i + 1) → ( ζ, i ) • Γ T is the quotient of k T by the ideals generated by the following relations(4.35) a r = 0 m = r, m ′ = ra u = ζ − z (( γ ) r ζ − ( γ ′ ) r ) m = m ′ = ra u = ζ z γ r m = r, m ′ = ra u = ( − r − d +1 ζ − z ( γ ′ ) r m = r, m ′ = r Note that the connected components of T have the form of (4.3) without the loops on eachvertex.Let V ∈ B T and Φ( V ) = W = ( W λ,i , A λ,i , B λ,i ). Note that m and m ′ must divide n by Corollary3.9, so g ℓ has ( n/ℓ ) th roots of unity as eigenvalues and is diagonalizable. Therefore, the B λ,i mapsare 0. At this point, we separate our analysis into two cases. Case 1: m = r and m ′ = r . In this case, Corollary 3.9 implies that γ = 0 and γ ′ = 0. Since the A λ,i maps are restrictions of σ , it follows from Corollary 3.24 that Φ( V ) is a representation of Γ T .Note that in this case, Γ T is the quotient of k T by ( k T ) r = rad r ( k T ). Case 2: m = r or m ′ = r . In this case, d divides r and r divides ℓ , so q ℓ = 1, and T has n/ℓ connected components of length d . A connected component of T is shown below where ζ is an( n/ℓ ) th root of unity.(4.36) ( ζ, d −
1) ( ζ, d − · · · ( ζ, a ζ,d − a ζ,d − a ζ, a ζ,d − It follows from Corollary 3.24, and the fact that kz m is always a multiple of ℓ , that Φ( V ) satisfiesthe relations necessary to be a representation of Γ T . Remark 4.37.
Suppose we are in Case 2, and that r = n , so T ( r, n ) = T ( n ) is the n th Taft algebra.Then the reductions above (and switching m, m ′ if necessary) imply that the setup simplifies to d = m ≤ m ′ = n = ℓ . Thus q ℓ = 1 and T has a single connected component consisting of thevertices (1 , i ) where 0 ≤ i < m . (cid:3) Both cases above result in the following corollary.
Corollary 4.38.
The category B T is equivalent to rep (Γ T ) . Proof.
We have shown above that Φ restricts to a functor from B T to rep (Γ T ), so we need to showthat each isomorphism class in rep (Γ T ) is in the image of this functor.Given a representation W of Γ T , we can see this a representation in N with all B λ,i = 0. Thus,Ψ( W ) is in B . It follows from (3.18) that given the action of x on k G ⊗ H ˜ W defined in (4.23), themap σ from Theorem 3.13 is given by(4.39) σ ( g t ⊗ w ) = q − t g t + ǫ ⊗ A λ,k ( w )for t ∈ Z + and w ∈ W ζ,i . Using this to compute the r th power of σ , we see that Corollary 3.24implies that Ψ( W ) ∈ B T . (cid:3) It is easy to see that Γ T is always finite dimensional over k . When the relations defining Γ T onlyset paths to 0, it is a self-injective Nakayama algebra. The indecomposable objects in this situationcan easily be described explicitly [ASS06, Ch. V], resulting in the following. Corollary 4.40.
In Case 1 and in each case where the scalar on the right hand side of (4.35) vanishes, the bimodule category B T is uniserial. This raises the following natural questions.
Question 4.41.
When the scalars on the right hand sides of the equations in (4.35) are nonzero,is rep (Γ T ) semisimple? If not, what is a quiver with admissible relations giving a Morita equivalentalgebra? (cid:3) Bimodules for U q ( sl ) and u q ( sl ) . Finally, we extend our results to U q ( sl ) and u q ( sl )by gluing over Borel subalgebras the equivalences from previous sections. We will make somesimplifying assumptions to avoid degenerate cases; see Notation 4.43 below. First, we introducethe relevant quivers and algebras.Fix a root of unity µ , a constant η ∈ k × and a positive integer N . Define a quiver Q ′ ( µ, N ) withthe same vertex set and arrows as Q ( µ, N ) along with additional c -type arrows ( λ, k ) → ( µλ, k − λ, k ).Denote the connected components of this quiver by S ′ p where p = lcm ( | µ | , N ). Each vertex( λ, k ) has an a -type arrow and a c -type arrow coming out of it. Then, ( λ, k ) is contained in a cycleof the form a p and another cycle of the form c p . It follows that S ′ p has p vertices with p cyclesconsisting of a -type arrows and p cycles consisting of c -type arrows.(4.42) S ′ = ( µ λ, k + 2) ( µλ, k + 1) ( λ, k )( λ, k + 1) ( µ λ, k ) ( µλ, k − µλ, k ) ( λ, k −
1) ( µ λ, k − b b bb b bb b ba aa aa aa a ac c cc c cc c c We assume that m, m ′ > q is a root of unity. For simplicity, we restrict to the OPF ACTIONS OF SOME QUANTUM GROUPS ON PATH ALGEBRAS 21 case that q is a primitive n th root of unity with n > n odd, which by Corollary 3.35 forcesthat both m, m ′ are divisible by n . Furthermore, this implies that the hypothesis of Remark 3.40is satisfied. Notation 4.43.
We fix the following notation. • Γ ′ ( µ, η, N ) is the quotient of k Q ′ ( µ, N ) by the ideal generated by all relations of the form(4.44) ab = µ − ba, cb = µbc, ηac = ca. • N ′ := nlrep Γ ′ ( q ℓ , q , d ) • S m and S m ′ are defined the same way as in Section 4.2 except we assume these algebrashave fixed U q ( sl )-actions extending the G -actions • B ′ is the category of finite dimensional S m - S m ′ -bimodules in rep ( U q ( sl ))Denote elements of N ′ by W = ( W λ,k , A λ,k , C λ,k , B λ,k ) where(4.45) A λ,k ∈ Hom k ( W λ,k , W µλ,k +1 ) , C λ,k ∈ Hom k ( W λ,k , W µ − λ,k +1 ) , B λ,k ∈ End k ( W λ,k ) . Theorem 4.46.
There exist mutually quasi-inverse functors (4.47) Φ ′ : B ′ → N ′ and Ψ ′ : N ′ → B ′ , and therefore the categories B ′ and N ′ are equivalent.Proof. Let G = h K i and fix a bimodule V ∈ B ′ . The bimodule V is a representation of both theBorel subalgebras U q ( b ) and U q − ( b ) as in (2.7). By Proposition 4.18, we have a functor Φ E sothat Φ E ( V ) is a representation of Γ( q ℓ , d ) denoted ( W λ,k , A λ,k , B λ,k ) and a functor Φ F so thatΦ F ( V ) is a representation of Γ( q − ℓ , d ) denoted ( W λ,k , C λ,k , B λ,k ). It follows from (3.41) that(4.48) C q ℓ λ,k +1 A λ,k = q A q − ℓ λ,k +1 C λ,k , and therefore ( W λ,k , A λ,k , C λ,k , B λ,k ) ∈ N ′ . Let Φ ′ ( V ) = ( W λ,k , A λ,k , C λ,k , B λ,k ).Notice that Γ ′ ( q ℓ , q , d ) contains subalgebras Γ( q ℓ , d ) and Γ( q − ℓ,d ) generated by the sets ofarrows { a λ,k , b λ,k } and { b λ,k , c λ, } respectively. For a representation W = ( W λ,k , A λ,k , C λ,k , B λ,k ) in N ′ , the subrepresentation W ′ = ( W λ,k , A λ,k , B λ,k ) is a representation of Γ( q ℓ , d ) and the subrepre-sentation W ′′ = ( W λ,k , C λ,k , B λ,k ) is a representation of Γ( q − ℓ,d ). We can construct the space ˜ W from the spaces W λ,k as in (4.21) and (4.22).By Proposition 4.25, there exists a functor Ψ E so that Ψ E ( W ′ ) = k G ⊗ H ˜ W is an S m - S m ′ -bimodule in rep ( U q ( b )) and there exists a functor Ψ F so that Ψ( W ′′ ) = k G ⊗ H ˜ W is an S m - S m ′ -bimodule in rep ( U q − ( b )). By construction of the two functors Φ E and Φ F , the S m - S m ′ -bimodulestructures on Ψ E ( W ′ ) and Ψ F ( W ′′ ) are identical. The actions of K on k G ⊗ ˜ W given by Ψ E andΨ F are also identical since they are induced by the same set of maps B λ,k .Let { γ Ei } m − i =0 and { γ E ′ j } m ′ − j =0 be the scalars from Proposition 3.28 determining the actions of thegenerator E of U q ( sl ) on S m and S m ′ respectively. For w ∈ W λ,k and t ∈ Z , letting j = τ ( λ, k ),we define an action of E on k G ⊗ ˜ W by(4.49) E · ( K t ⊗ w ) = q − t (cid:16) γ E ′ j K t ⊗ w − γ E q − K t +1 ⊗ w + K t + ǫ ⊗ A λ,k ( w ) (cid:17) . Similarly, let { γ Fi } m − i =0 and { γ F ′ j } m ′ − j =0 be the scalars determining the action of the generator F of U q ( sl ) on S m and S m ′ respectively. We define an action of F on k G ⊗ ˜ W by(4.50) F · ( K t ⊗ w ) = q − t (cid:16) γ F ′ j K t − ⊗ w − γ F q − K t ⊗ w + K t − ǫ ⊗ C λ,k ( w ) (cid:17) . Notice that the actions defined in (4.49) and (4.50) are exactly the actions of the generators x of the two Borel subalgebras of U q ( sl ) on k G ⊗ ˜ W given by the functors Ψ E and Ψ F under the identifications in (2.7). One can check that these action satisfy relations (2.5), so k G ⊗ H ˜ W ∈ B ′ .Let Ψ ′ ( W ) = k G ⊗ H ˜ W .The fact that Φ ′ and Ψ ′ are mutually quasi-inverse follows directly from Theorem 4.26 whichgives us that Φ E , Ψ E and Φ F , Ψ F are both pairs of mutually quasi-inverse functors. (cid:3) Now, we consider the bimodules in B ′ whose U q ( sl ) actions factor through u q ( sl ). From Corol-lary 3.34 and our simplifying assumption that m, m ′ >
2, we are reduced to the case m = m ′ = n .Then this makes ℓ = d = n and thus q ℓ = 1. Notation 4.51.
We fix the following notation • B ′ T is the full subcategory of bimodules in B ′ whose U q ( sl ) actions factor through u q ( sl ). • T ′ is the quiver(4.52) T ′ = (1 , d −
1) (1 , d − · · · (1 , a a aac c cc • Γ ′ T is the quotient of k T ′ generated by all relations of the form(4.53) q ac = ca, a d = ( γ E ) n − ( γ E ′ ) n , and c d = ( γ F ) n − ( γ F ′ ) n . Corollary 4.54.
The categories B ′ T and rep (Γ ′ T ) are equivalent.Proof. Recall that the two Borel subalgebras of u q ( sl ) are Taft algebras. Since we are assuming m and m ′ are greater than 2, it follows from Corollary 3.35 that m = m ′ = r = n , and Remark 4.37applies to these Borel subalgebras. The statement now follows from Theorem 4.46 and Corollary4.38. (cid:3) A standard reordering argument shows that Γ ′ T is always finite dimensional over k . This raisesthe following natural questions. Question 4.55.
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University of Iowa, Department of Mathematics, Iowa City, USA
Email address , Ryan Kinser: [email protected]
University of Iowa, Department of Mathematics, Iowa City, USA
Email address , Amrei Oswald:, Amrei Oswald: