aa r X i v : . [ m a t h . QA ] A ug GENERALISED TAFT ALGEBRAS AND PAIRS ININVOLUTION
SEBASTIAN HALBIG
Technische Universitt Dresden, Institut f¨ur Geometrie,Zellescher Weg 12-14, 01062 Dresden
Abstract.
A class of finite-dimensional Hopf algebras which generalise thenotion of Taft algebras is studied. We give necessary and sufficient conditionsfor these Hopf algebras to omit a pair in involution, that is, to not have agroup-like and a character implementing the square of the antipode. As aconsequence we prove the existence of an infinite set of examples of finite-dimensional Hopf algebras without such pairs. This has implications for thetheory of anti-Yetter-Drinfeld modules as well as biduality of representationsof Hopf algebras.
Acknowledgements. The author would like to thank P. Hajac for his kind invitationto IMPAN. He would also like to thank U. Kr¨ahmer for many fruitful discussions.1.
Introduction
Main result.
A pair in involution for a Hopf algebra H over a field k is atuple ( l, β ) of a group-like l ∈ H and a character β : H → k such that the squareof the antipode is given by the conjugate action of l and β . In a vague sense itcan be imagined to be similar to a ‘square-root’ of Radford’s S -formula. Oftenone additionally requires the pair to be modular, i.e to satisfy β ( l ) = 1. Pairs ininvolution appear in many different contexts within Hopf algebra theory, reachingfrom Hopf-cyclic cohomology [3, 5, 4] to knot invariants [9]. Kauffman and Radfordshowed in the aforementioned paper that for some Taft algebras the square of theantipode is not implemented by the square-roots of the distinguished group-likes, see[9, Proposition 7]. They however still have modular pairs in involution. The aim ofthe present paper is to prove that there are finite-dimensional Hopf algebras withoutpairs in involution. In a previous work by Kr¨ahmer and the author, see [6], theexistence of finite-dimensional Hopf algebras without modular pairs in involutionwas shown. The Hopf algebras studied therein are in fact examples of a wider classwe will refer to as generalised Taft algebras. As algebras, these are generated by agroup-like g and two twisted primitives x , y such that g spans a cyclic group of order N and x , y are nilpotent. Moreover the generators are required to commute up tosome N -th roots of unity. The precise definition of this presentation can be foundin Theorem 5. For a generalised Taft algebra a pair in involution corresponds E-mail address : [email protected] . Date : August 29, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Hopf algebra, pair in involution, pivotal category, anti-Drinfeld double. to a solution of a system of Diophantine equations, see Theorem 7. The mainresult, Theorem 8, gives necessary and sufficient conditions for the non-existenceof such solutions. As an application we prove that there are finite-dimensionalHopf algebras without these pairs, see Lemma 7. In Lemma 10 we show that Hopfalgebras, which are isomorphic as algebras, need not share the property of havinga pair in involution.1.2.
Pivotal categories and anti-Yetter-Drinfeld modules.
A pivotal cate-gory is a monoidal category with a notion of duality and a natural isomorphismbetween any object and its bidual which is compatible with the monoidal structure.One example are finite-dimensional modules over pivotal Hopf algebras. For a finite-dimensional Hopf algebra having a pair in involution is equivalent to its Drinfelddouble being pivotal. Our results imply that the finite-dimensional modules of theDrinfeld double of a Hopf algebra need not form a pivotal category.Anti-Yetter-Drinfeld modules arose in the field of Hopf-cyclic cohomology. Sim-ilar to Yetter-Drinfeld modules, they are simultaneously modules and comodulesover a Hopf algebra with a compatibility condition between the action and coac-tion. For a finite-dimensional Hopf algebra one can construct an algebra called theanti-Drinfeld double whose modules correspond to anti-Yetter-Drinfeld modules.The existence of a pair in involution is equivalent to the existence of an isomor-phism of algebras between the Drinfeld double and the anti-Drinfeld double. Asa consequence of our findings there are examples where such an isomorphism doesnot exist.1.3.
Outline.
This paper is organized as follows. Section 2 serves as a summary ofthe theory of pairs in involution with a focus on finite-dimensional Hopf algebras.In Section 3 we introduce the main object of our study - generalised Taft algebras- and give an ‘easy-to-work-with’ presentation. The classification of all generalisedTaft algebras that omit a pair in involution is carried out in Section 4. It is achievedin two steps. First we show that the existence of a pair in involution correspondsto the solvability of a system of Diophantine equations. Then we classify all suchsystems that do not have a solution. We conclude the paper with Section 5, wherewe apply our results to the context of representation theory.2.
Pairs in involution
We work over an algebraically closed field k of characteristic zero; ‘dim’ and ‘ ⊗ ’ought to be understood as dimension and tensor product over k . Standard notationfor Hopf algebras, as in e.g. [10, 12], is freely used. Given a Hopf algebra H we write Gr ( H ) for its group of group-likes, P r ( H ) for its space of primitive elements and H ◦ for its (finite) dual Hopf algebra. The antipode of H is denoted by S : H → H ,its counit by ǫ : H → k and its coproduct by ∆ : H → H ⊗ H . For calculationsinvolving the coproduct of H or the coaction of some H comodule M we rely onreduced Sweedler notation. For example we write h (1) ⊗ h (2) := ∆( h ) for h ∈ H .An element x ∈ H whose coproduct is ∆( x ) = 1 ⊗ x + x ⊗ g , with g ∈ Gr ( H ) iscalled a twisted primitive.Modular pairs in involution play the role of coefficients for Hopf-cyclic cohomol-ogy as introduced by Connes and Moscivici [3]. Later on it was realised by Hajac ENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION 3 et. al. that this notion can be extended to that of (stable) anti-Yetter-Drinfeldmodules, see Section 3.
Definition 1.
Let H be a Hopf algebra over k . A tuple ( l, β ) comprising group-like elements l ∈ Gr ( H ) and β ∈ Gr ( H ◦ ) is a pair in involution if it satisfies theantipode condition(AC) S ( h ) = β ( h (3) ) β − ( h (1) ) lh (2) l − , for all h ∈ H. If additionally the modularity condition(MC) β ( l ) = 1holds, it is called a modular pair in involution.A left integral of a Hopf algebra H is an element Λ ∈ H such that h Λ = ǫ ( h )Λ forall h ∈ H . If H is finite-dimensional left integrals form a one-dimensional subspace L ( H ) ⊂ H and there is a unique group-like α ∈ Gr ( H ◦ ) such that Λ h = α ( h )Λ forall Λ ∈ L ( H ) and h ∈ H . It is called the distinguished group-like of H . Theorem 1 (Radford’s S -formula) . Let H be a finite-dimensional Hopf algebraand g ∈ Gr ( H ) , α ∈ Gr ( H ◦ ) the distinguished group-likes of H ◦ and H . Then thefourth power of the antipode is given by (1) S ( h ) = α ( h ) α − ( h (1) ) g − h (2) g, for all h ∈ H. In their paper on the classification of ribbon elements of Drinfeld doubles, see[9], Kauffman and Radford studied ‘square roots’ of the distinguished group-likes toobtain a formula for the square of the antipode. As an implication of [9, Proposition6] we obtain:
Lemma 1.
Let H be a pointed Hopf algebra, i.e. a Hopf algebra whose simplecomodules are one-dimensional. If the dimension of H is odd it has a pair ininvolution ( l, β ) such that g := l and α := β are the distinguished group-likes of H ◦ and H respectively. Let us conclude this section with a remark on the representation theoretic view-point on pairs in involution. Given a finite-dimensional Hopf algebra H on canassociate to it its category of finite-dimensional Yetter-Drinfeld modules, see Sec-tion 3.1. It is a rigid category; i.e it is monoidal together with a notion of dualitywhich is compatible with its monoidal structure. In this context a pair in involu-tion corresponds to and can be reconstructed from a monoidal natural isomorphismbetween the identity functor and the functor that maps objects and morphisms totheir bidual. A category with such a structure is called a pivotal category.3. Generalised Taft algebras
The strategy behind defining generalised Taft algebras is as follows. Fix a finitecyclic group G . Choose a Yetter-Drinfeld module V over the group algebra kG whose braiding σ : V ⊗ V → V ⊗ V is subject to certain relations. The bosonisationof the Nichols algebra B ( V ) of V along kG yields a Hopf algebra structure on thevector space B ( V ) ⊗ kG . This will be referred to as the coopposite of a generalisedTaft algebra, see Definition 5. A presentation in terms of generators and relationsis given in Theorem 5.An extensive survey on the subject of Nichols algebras can be found in [1, 2]. GENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION
Yetter-Drinfeld and anti-Yetter-Drinfeld modules.
Every Hopf algebrain this section is assumed to have an invertible antipode.
Definition 2. A Yetter-Drinfeld module over a Hopf algebra H is a k -vector space M together with a module structure ⊲ : H ⊗ M → M and a comodule structure δ : M → H ⊗ M such that(YD) δ ( h ⊲ m ) = h (1) m ( − S ( h (3) ) ⊗ h (2) ⊲ m (0) , ∀ h ∈ H, m ∈ M. A linear map f : M → N between Yetter-Drinfeld modules is called a morphism ofYetter-Drinfeld modules if it is a module and comodule morphism. Remark.
The Yetter-Drinfeld modules over a Hopf algebra H form the category HH YD . The diagonal action and coaction of H define a monoidal structure on it. Ifthe antipode of H is invertible HH YD is braided monoidal with braiding(2) σ M,N : M ⊗ N → N ⊗ M, m ⊗ n m ( − ⊲ n ⊗ m (0) . Yetter-Drinfeld modules over a finite-dimensional Hopf algebra H coincide withmodules over its Drinfeld double D ( H ). Recall that D ( H ) is the vector space H ◦ ⊗ H whose Hopf algebra structure is defined by( α ⊗ g )( β ⊗ h ) := α (1) ( h (1) ) α (3) ( S ( h (3) )) βα (2) ⊗ gh (2) , ∆( α ⊗ g ) := ( α (1) ⊗ g (1) ) ⊗ ( α (2) ⊗ g (2) ) ,S ( α ⊗ g ) := α (1) ( S ( g (1) )) α (3) ( g (3) ) S − ( α (2) ) ⊗ S ( g (2) ) . Replacing the antipode with its inverse in the definition of the multiplication yieldsanother algebra, the anti-Drinfeld double A ( H ). Namely the vector space H ◦ ⊗ H together with the multiplication( α ⊗ g )( β ⊗ h ) := α (1) ( h (1) ) α (3) ( S − ( h (3) )) βα (2) ⊗ gh (2) . Modules over A ( H ) correspond to anti-Yetter-Drinfeld modules . That is, triples( M, ⊲, δ ) comprising a vector space M , an action ⊲ : H ⊗ M → M , a coaction δ : M → H ⊗ M and the compatibility condition(AYD) δ ( h ⊲ m ) = h (1) m ( − S − ( h (3) ) ⊗ h (2) ⊲ m (0) , ∀ h ∈ H, m ∈ M. One-dimensional anti-Yetter-Drinfeld modules are the same as pairs in involution.In general anti-Yetter-Drinfeld modules do not form a monoidal category but amodule category over the Yetter-Drinfeld modules. The following is an unpublishedresult by Hajac and Sommerh¨auser. For the readers convenience we include a proof.
Theorem 2.
Suppose H is a finite-dimensional Hopf algebra. Then the anti-Drinfeld double A ( H ) is isomorphic to the Drinfeld double D ( H ) as an algebraif and only if H has a pair in involution.Proof. Assume f : A ( H ) → D ( H ) to be an isomorphism of algebras. The trivial D ( H ) module ǫ k becomes an anti-Yetter-Drinfeld module ( k, δ, ⊲ ) by pulling backthe action along f . This implies the existence of a pair of group-like elements β − ∈ Gr ( H ◦ ) and l ∈ Gr ( H ) implementing the action and coaction. We write δ ′ : k → H , λ λ (0) λ ( − . For h ∈ H we have( β − ( h (2) ) S ( h (1) ) l = S ( h (1) ) δ ′ ( h (2) ⊲ (AYD) = β − ( h (1) ) lS − ( h (2) ) . Applying S to both sides proves this to be equivalent to ( l, β ) being a pair ininvolution. ENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION 5
Conversely let ( l, β ) be a pair in involution and define f : A ( H ) → D ( H ), α ⊗ g α (2) ( l ) β − ( g (2) ) α (1) ⊗ g (1) . The following computation shows that it is amorphism of algebras: f (( α ⊗ g )( γ ⊗ h )) = α (1) ( h (1) ) α (3) ( S − ( h (3) )) f ( γα (2) ⊗ gh (2) )= α (1) ( h (1) ) α (4) ( S − ( h (4) )) γ (2) ( l ) α (3) ( l ) β − ( g (2) ) β − ( h (3) ) γ (1) α (2) ⊗ g (1) h (2) = α (1) ( h (1) ) α (3) ( β − ( h (3) ) lS − ( h (4) )) γ (2) ( l ) β − ( g (2) ) γ (1) α (2) ⊗ g (1) h (2)( ) = α (1) ( h (1) ) α (3) ( S ( h (3) )) α (4) ( l ) γ (2) ( l ) β − ( g (2) ) β − ( h (4) ) γ (1) α (2) ⊗ g (1) h (2) = f (( α ⊗ g )) f (( γ ⊗ h )) . Its inverse is given by f − : D ( H ) → A ( H ), α ⊗ g α (2) ( l − ) β ( g (2) ) α (1) ⊗ g (1) . (cid:3) As it turns out the existence of a pair in involution corresponds to a suitablystrong notion of Morita equivalence between the Drinfeld and anti-Drinfeld double.Given an algebra A we write F org A : A - Bimod → A - M od for the forgetful functorfrom the category of bimodules over A to the category of A -modules. Lemma 2.
Let H be a finite-dimensional Hopf algebra. It is equivalent: (1) H has a pair in involution. (2) There are k -linear equivalences of categories F : D ( H ) - M od → A ( H ) - M od and G : D ( H ) - Bimod → A ( H ) - Bimod such that a natural isomorphism η : F Forg D ( H ) → Forg A ( H ) G exists.Proof. Suppose H has a pair in involution. By Theorem 2 there exists an isomor-phism of algebras f : A ( H ) → D ( H ). Let F : D ( H )- M od → A ( H )- M od be thefunctor that identifies D ( H )-modules with A ( H )-modules by pulling back the ac-tion along f and define G : D ( H )- Bimod → A ( H )- Bimod likewise. Both, F and G , are k -linear equivalences of categories and F Forg D ( H ) = Forg A ( H ) G .Conversely assume F , G and η to be as described above. Let X bi := ( X, ⊲, ⊳ ) bea D ( H )-bimodule. Set X l := Forg D ( H ) ( X bi ) = ( X, ⊲ ) and write Y := F ( X l ). The A ( H )-module endomorphisms End A ( H ) ( Y ) themselves become an A ( H )-module by˜ ⊲ : A ( H ) ⊗ End A ( H ) ( Y ) → End A ( H ) ( Y ) , ( a ˜ ⊲φ )( x ) := φ ( η − X bi ( η X bi ( x ) ⊳ a )) . As F is a k -linear equivalence of categories End D ( H ) ( X l ) ∼ = End A ( H ) ( Y ) as k -vectorspaces. Choose X bi = ǫ k ǫ to be the trivial D ( H )-bimodule. Then X l = ǫ k is thetrivial D ( H )-module and End A ( H ) ( F ( ǫ k )) is a one-dimensional module over theanti-Drinfeld double A ( H ), i.e. a pair in involution. (cid:3) Nichols algebras and bosonisations.
The definition of a Hopf algebra gen-eralises naturally to braided monoidal categories. A Hopf algebra R in the categoryof Yetter-Drinfeld modules over some Hopf algebra H is referred to as a braided Hopf algebra (over H ). Theorem 3.
Let V be a Yetter-Drinfeld module over a Hopf algebra H with in-vertible antipode. The Nichols algebra of V is the braided graded Hopf algebra B ( V ) = ⊕ n ≥ B ( V )( n ) which is up to unique isomorphism uniquely determined by (1) B ( V )(0) = k , (2) B ( V )(1) = P r ( B ( V )) = V and (3) B ( V ) is generated as an algebra by V . Different types of Nichols algebras are distinguished in terms of their braidings.
GENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION
Definition 3.
Let V be a Yetter-Drinfeld module of k -dimension θ over a Hopfalgebra H and σ := σ V,V : V ⊗ V → V ⊗ V its braiding. An ordered k -basis v , . . . v θ of V is said to be of diagonal type if(3) σ ( v i ⊗ v j ) = q ij v j ⊗ v i , q ij ∈ k, ≤ i, j ≤ θ. The matrix ( q ij ) ≤ i,j ≤ θ is called the matrix of the braiding . Accordingly V and B ( V ) are referred to be of diagonal type if V has an ordered basis of diagonal type.Finite-dimensional Nichols algebras of diagonal type were classified by Hecken-berger in terms of generalised Dynkin diagrams [8]. The Nichols algebra part of ageneralised Taft algebra corresponds to a diagram of A × A ∼ = D -type. Definition 4.
Let H be a Hopf algebra with invertible antipode. A Yetter-Drinfeldmodule V over H is of D -type if it has an ordered k -basis { v , v } of diagonal typeand the matrix of the braiding ( q ij ) ≤ i,j ≤ satisfies:(4) Its entries are roots of unity, q , q = 1 and q q = 1 . Its Nichols algebra B ( V ) is accordingly also referred to as of D -type.The bosonisation of a braided Hopf algebra R over H equips the vector space R ⊗ H with the structure of a Hopf algebra. To distinguish between the coactionand comultiplication of R we use a slight variation of Sweedler notation and write r (1) ⊗ r (2) := ∆( r ) for r ∈ R . Theorem 4.
Let H be a Hopf algebra with invertible antipode and R a braidedHopf algebra in HH YD . The bosonisation of R by H is the Hopf algebra R H ,whose underling vector space is R ⊗ H and whose multiplication, comultiplicationand antipode is defined by (5) ( r ⊗ g )( s ⊗ h ) := r ( g (1) ⊲ s ) ⊗ g (2) h, ∆( r ⊗ g ) := r (1) ⊗ ( r (2) ) ( − g (1) ⊗ ( r (2) ) (0) ⊗ g (2) ,S ( r ⊗ g ) := S H ( r ( − g (2) ) ⊲ S R ( r (0) ) ⊗ S H ( r ( − g (1) ) . Definition 5.
Let V be a Yetter-Drinfeld module of D -type over the group algebra kG of a finite cyclic group G . We call ( B ( V ) kG ) cop a generalised Taft algebra .The identification of N -th roots of unity with Z N allows us to characterise theNichols algebra part of a generalised Taft algebra in terms of an ordered set of in-tegers modulo N subject to certain (in-)equations. The Pontryagin dual of a finiteabelian group G is denoted by b G := { ξ : G → k × | ξ is a homomorphism of groups } . Lemma 3.
Let N ≥ and ( V, ⊲, δ ) be a two-dimensional Yetter-Drinfeld moduleover the group algebra k Z N . The following are equivalent: (1) V is of D -type. (2) For any pair of generators g ∈ Z N and ξ ∈ b Z N there exists a k -basis { x, y } of V and integers a , a , b , b ∈ Z N such that g⊲x = ξ b ( g ) x , g⊲y = ξ b ( g ) y and δ ( x ) = g a ⊗ x , δ ( y ) = g a ⊗ y with (6) a b = 0 , a b = 0 , a b + a b = 0 . Proof.
Let V be of D -type. Fix generators g ∈ Z N and ξ ∈ b Z N of their respectivegroups and define q := ξ ( g ). As Yetter-Drinfeld modules V ∼ = ⊕ i,j ∈ Z N V ξ j g i , where V ξ j g i := { v ∈ V | h ⊲ v = ξ j ( h ) v and δ ( v ) = g i ⊗ v } . It follows that a homogenous
ENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION 7 ordered k -basis { x, y } of V exists whose matrix of the braiding satisfies (4). Thatis, there are integers a , a , b , b ∈ Z N with x ∈ V ξ b g a and y ∈ V ξ b g a and, under theidentification { q n ∈ k | ≤ n < N } ∼ = Z N , q
1, (4) translates to (6).Fix generators g ∈ Z N , ξ ∈ b Z N , of Z N and b Z N respectively and set q := ξ ( g ). Let { x, y } ⊂ V be a basis with a , a , b , b ∈ Z N as above. Then { x, y } is of diagonaltype with its matrix of the braiding ( q ij ) ≤ i,j ≤ given by q ij = q a i b j . Identifying Z N ∼ = { q n ∈ k | ≤ n < N } , 1 q shows that (6) implies (4). (cid:3) Definition 6.
Let N ≥
2. A tuple ( a , a , b , b ) ∈ Z N whose elements satisfy (6)is called a tuple of parameters of a generalised Taft algebra .From a presentation of Nichols algebras of D -type we derive a presentation ofgeneralised Taft algebras. By [7, Corollary 15] we have: Lemma 4.
Let V be a Yetter-Drinfeld module over a Hopf algebra H and { v , v } a basis of diagonal type whose matrix of the braiding ( q ij ) ≤ i,j ≤ satisfies (4) . Set m = min i ∈ N ( q i = 1) and n = min i ∈ N ( q i = 1) . The Nichols algebra B ( V ) ispresented by the generators v , v and relations v m = 0 , v n = 0 , v v = q v v . The set { v i v j | ≤ i < m, ≤ j < n } is a k -basis of B ( V ) . We write |h a i| for the order of an element a ∈ G of a group G . Theorem 5.
Let N ≥ , q ∈ k a primitive N -th root of unity and a , a , b , b ∈ Z N satisfying (6) . Set N x := |h a b i| and N y := |h a b i| . Define H := H q ( a , a , b , b ) to be the Hopf algebra generated by the elements g , x and y and relations g N = 1 , x N x = 0 , y N y = 0 ,gx = q b xg, gy = q b yg, xy = q a b yx, ∆( g ) = g ⊗ g, ∆( x ) = 1 ⊗ x + x ⊗ g a , ∆( y ) = 1 ⊗ y + y ⊗ g a ,S ( g ) = g − , S ( x ) = − xg − a , S ( y ) = − yg − a . Then H is isomorphic to a generalised Taft algebra and every generalised Taftalgebra is of the above form.Proof. Let V be a Yetter-Drinfeld module of D -type over k Z N . Fix generators h ∈ Z N and ξ ∈ c Z N and write q := ξ ( h ). By Lemma 3 there exists a basis { v , v } of V and parameters of a generalised Taft algebra a , a , b , b ∈ Z N such that h ⊲ v i = q b i v i and δ ( v i ) = h a i ⊗ v i for 1 ≤ i ≤
2. The generalised Taft algebra( B ( V ) k Z N ) cop is generated by the elements g := 1 ⊗ h , x := v ⊗ y := v ⊗ D -type given in Lemma 4 and the Hopfalgebra structure of bosonisations imply that g , x and y are subject to preciselythe above relations.The converse follows by the fact that for N ≥ a , a , b , b ∈ Z N and a primitive N -th root of unity q ∈ k determines a Yetter-Drinfeld module V of D -type over k Z N . By the aboveargument ( B ( V ) k Z N ) cop has the desired presentation. (cid:3) Convention.
We fix the notation of Theorem 5. From now onwards N ≥ = q ∈ k a primitive N -th root of unity. Given a generalisedTaft algebra H q ( a , a , b , b ) we write g , x and y for its generators. In particular GENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION g generates the group Gr ( H ) and x and y are nilpotent of degree N x and N y respectively. 4. Hopf algebras without pairs in involution
We show that, similar to Taft algebras, their generalisations are a class of basic,pointed Hopf algebras, which is closed under duality. Our main result, Theorem 8,states necessary and sufficient conditions for a generalised Taft algebra to omit apair in involution. Thereafter we investigate how various properties of these Hopfalgebras effect the existence of such pairs. In particular we construct an infinitefamily of examples of finite-dimensional Hopf algebras without pairs in involution,see Lemma 7.4.1.
Properties of generalised Taft algebras.
To prove that the class of gen-eralised Taft algebras is closed under duality we identify a set of generators of thedual. Let H := H q ( a , a , b , b ) be a generalised Taft algebra with generators g , x and y . Set ξ, ψ, φ : H → k ,(7) ξ ( x i y j g l ) = q − l δ i = j =0 , ψ ( x i y j g l ) = q − b l δ i =1 ,j =0 ,φ ( x i y j g l ) = q − b l δ i =0 ,j =1 . Theorem 6.
Let H := H q ( a , a , b , b ) be a generalised Taft algebra. Write ¯ g , ¯ x and ¯ y for the generators of H q ( b , b , a , a ) . The map Θ : H q ( b , b , a , a ) → H ◦ , Θ(¯ g ) = ξ , Θ(¯ x ) = ψ and Θ(¯ y ) = φ is an isomorphism of Hopf algebras.Proof. Let g , x , y denote the generators of H , with g N = 1 and x N x = y N y = 0.Evaluating ξ , φ and ψ on the basis { x i y j g l | ≤ i, j, l ≤ N x , N y , N } of H yields ξ N = ǫ, ψ N x = 0 , φ N y = 0 ,ξψ = q a ψξ, ξφ = q a φξ, ψφ = q b a φψ and ∆( ξ ) = ξ ⊗ ξ, ∆( ψ ) = ǫ ⊗ ψ + ψ ⊗ ξ b , ∆( φ ) = ǫ ⊗ φ + φ ⊗ ξ b . Therefore Θ is a well-defined homomorphism of Hopf algebras. A direct computa-tion shows that( ξ r ψ s φ t ) (cid:18)(cid:18)X N − m =0 q i · m g m (cid:19) x j y l (cid:19) = 0 ⇔ i = r, j = s, l = r. Thus { ξ r ψ s φ t | ≤ r, s, t ≤ N, N x , N y } ⊂ im Θ is a linearly independent subset of H ◦ whose cardinality is dim H ◦ , implying that Θ is an isomorphism. (cid:3) A Hopf algebra is called pointed if every simple comodule is one-dimensional; ifevery simple module is one-dimensional it is called basic . Corollary 1.
Generalised Taft algebras are pointed and basic.Proof.
The first claim follows from the fact that generalised Taft algebras are gen-erated by group-likes and twisted primitives. The second is a consequence of theprevious observation and Theorem 6. (cid:3)
We determine the left-integrals and distinguished group-like of a generalised Taftalgebra. The latter will prove useful in the study of the square of the antipode.
ENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION 9
Lemma 5.
Let q ∈ k be a primitive N -th root of unity and H := H q ( a , a , b , b ) a generalised Taft algebra. Up to scalar multiplicatives the left integrals of H and H ◦ are Λ := (cid:18)X N − i =0 g i (cid:19) x N x − y N y − ∈ H and Υ := (cid:18)X N − i =0 ξ i (cid:19) ψ N x − φ N y − ∈ H ◦ . The distinguished group-likes of H and H ◦ are ξ − ( b + b ) and g − ( a + a ) .Proof. By multiplying Λ with the generators of H we see that it is a left integral andthat ξ b ( N x − b ( N y − is the distinguished group-like of H . As b N x = b N y = 0modulo N we have ξ b ( N x − b ( N y − = ξ − ( b + b ) . The results for H ◦ follow byapplying the isomorphism of Theorem 6. (cid:3) The motivation behind Kauffman’s and Radford’s study of the square of theantipode, see [9], was understanding Hopf algebras that give rise to knot invariants.This necessarily requires a quasitriangular structure on the Hopf algebra. That is,roughly speaking, an encoding of the notion of braiding on the level of Hopf algebras.
Lemma 6 ( [11, Theorem 3.4] ) . Let H := H q ( a , a , b , b ) be a generalised Taftalgebra with generators g , x , y and recall that N x , N y ∈ N were the minimal integerssuch that x N x = y N y = 0 . Then H has a quasitriangular structure if and only if N is divisible by , N x = N y = 2 and a = a = N . Pairs in involution as solutions of Diophantine equations.
To findnecessary and sufficient criteria for generalised Taft algebras to not admit a pair ininvolution we study the behaviour of the square of the antipode. Fix a generalisedTaft algebra H := H q ( a , a , b , b ). Its square of the antipode is determined by S ( g ) = g, S ( x ) = q a b x, S ( y ) = q a b y. Likewise the fourth power of the antipode is S ( g ) = g, S ( x ) = q a b x, S ( y ) = q a b y. Now consider the family of Hopf algebra automorphisms T ( l,β ) : H → H whichis indexed by a pair of group-likes l ∈ Gr ( H ), β ∈ Gr ( H ◦ ) and defined via T ( l,β ) ( h ) = β ( h (3) ) β − ( h (1) ) lh (2) l − , for all h ∈ H. Definition 1 states that H has a pair in involution if and only if there exists a pairof group-likes ( l, β ) such that T ( l,β ) = S . The cyclic groups Gr ( H ) and Gr ( H ◦ )are generated by g and ξ . Every group-like l ∈ Gr ( H ) and character β ∈ Gr ( H ◦ )can uniquely be written as l = g d and β = ξ − c , with c, d ∈ Z N . Evaluating T ( g d ,ξ − c ) on the generators yields T ( g d ,ξ − c ) ( g ) = g, T ( g d ,ξ − c ) ( x ) = q a c + b d x, T ( g d ,ξ − c ) ( y ) = q a c + b d y. Identifying { N -th roots of unity } → Z N , q Theorem 7.
Let q = 1 be a primitive N -th root of unity. A generalised Taft algebra H q ( a , a , b , b ) has a pair in involution if and only if there are c, d ∈ Z N such thatmodulo N (8) a c + b d = a b , a c + b d = a b . The pair is modular if and only if it additionally satisfies modulo N (9) cd = 0 . Remark.
Lemma 5 and the discussion prior to the theorem imply that for givenparameters a , a , b , b ∈ Z N of a generalised Taft algebra the integers c ′ := b + b ,and d ′ := a + a satisfy modulo Na c ′ + b d ′ = 2 a b , a c ′ + b d ′ = 2 a b . If N is odd 2 is invertible modulo N implying that (8) has a solution in that case.Consequently a generalised Taft algebra without pairs in involution necessarilyneeds to have a group of group-likes of even order N . In fact, the existence of sucha pair depends on the behaviour of the parameters modulo 2 n where n ∈ N is themaximal integer such that 2 n divides N . Definition 7.
Suppose 2 n · j ≥ j odd and let a , a , b , b ∈ Z n · j be theparameters of a generalised Taft algebra. If n > µ , a := a , µ , a := a , µ , b := b , µ , b := b ( mod 2 n ) , with µ i,j ∈ N such that µ i,j odd or zero for 1 ≤ i, j ≤ ≤ a , a , b , b < n minimal. We call µ := ( µ i,j ) ≤ i,j ≤ the coefficient matrix and a , a , b , b the powers of the parameters. The power of the coefficient matrix is the minimal integer τ ∈ N such that 2 τ ν = det µ mod 2 n with ν ∈ N odd or zero. Theorem 8.
Suppose n · j ≥ with j odd. Let q ∈ k be a primitive n j -th rootof unity and H := H q ( a , a , b , b ) be a generalised Taft algebra with powers of theparameters a , a , b , b , coefficient matrix µ := ( µ i,j ) ≤ i,j ≤ and power of thecoefficient matrix τ . It is equivalent: (1) H has no pair in involution. (2) n ≥ , µ i,j = 0 mod 2 n for ≤ i, j ≤ , a + b < n , a = a and τ > min { a , a } or det µ = 0 mod 2 n . Before giving a proof of the theorem we investigate the choice of a presentationof a generalised Taft algebra.
Remark.
Rearranging the parameters of H := H q ( a , a , b , b ) into ( a , a , b , b )amounts in the choice of another presentation of H . Assuming n ≥ µ ′ and τ ′ for the coefficient matrix and its power with regard to this new presentation.Observe that det µ = − det µ ′ implying that τ = τ ′ . If µ has only non-zero entriesmodulo 2 n the equation a b + a b = µ , µ , a + b + µ , µ , a + b = 0 mod 2 n implies a + b < n ⇔ a + b < n . Thus the criteria listed above are invariant underthis change of presentation and, whenever well-defined, we can assume a ≤ a . Proof.
Without loss of generality we assume that, whenever well-defined, a ≤ a .We prove that H having a pair in involution is equivalent to at least one of theabove conditions not being met. By Theorem 7 this amounts in finding ˜ c, ˜ d ∈ Z n · j such that modulo 2 n · ja ˜ c + b ˜ d = a b , a ˜ c + b ˜ d = a b . As χ : Z n · j → Z n × Z j , ( x mod 2 n · j ) ( x mod 2 n , x mod j ) is an isomorphismof rings this is equivalent to the existence of c ′ , d ′ ∈ Z n and c ′′ , d ′′ ∈ Z j such that a c ′ + b d ′ = a b , a c ′ + b d ′ = a b (mod 2 n ) , (10a) a c ′′ + b d ′′ = a b , a c ′′ + b d ′′ = a b , (mod j ) . (10b) ENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION 11
As shown in a previous remark c ′′ := 2 − ( b + b ) and d ′′ := 2 − ( a + a ) providesa solution to the second equation. Therefore H has a pair in involution if and onlyif a solution to (10a) exists. If n = 0 this is trivially the case. We assume n ≥ H be zero modulo 2 n , resulting in µ i,j = 0 modulo2 n for some 1 ≤ i, j ≤
2. By checking each of the four possible cases one observesthat there always exists a solution. For example if µ , = 0 take c ′ = b and d ′ = 0.Thus µ i,j = 0 from now onwards.Multiplying (10a) with ( µ , µ , ) − yields2 a c + µ b b d = µ b a + b (mod 2 n ) , (11a) µ a a c + 2 b d = µ a a + b (mod 2 n ) , (11b)with c := µ − , c ′ , d := µ − , d ′ and µ a := µ , µ − , , µ b := µ , µ − , . Since we assumed a ≤ a we can insert (11a) into (11b) and get modulo 2 n (12) µ a µ b a − a + b (2 a − d ) + 2 b d = µ a a + b ⇔ (2 b − µ a µ b a − a + b ) d = µ a a + b − µ a µ b a + b . Observe that since a b + a b = 0 mod N we have(13) 2 a + b + µ a µ b a + b = 0( mod 2 n ) . Thus (12) is equivalent to(14) (2 b − µ a µ b a + b − a ) d = 2 a + b ( µ a a − a + 1)( mod 2 n ) . If a + b ≥ n the right-hand side of (14) is zero and d = 0 a solution. In thefollowing let a + b < n .The equation (13) implies a + b = a + b , allowing us to rewrite (14) as(15) 2 b (1 − µ a µ b ) d = 2 a + b ( µ a a − a + 1)( mod 2 n ) . If a = a the right-hand side of (15) is given by 2 a + b +1 ρ for some integer ρ .The identity (13) implies that 1 + µ a µ b = 0 modulo 2 n − ( a + b ) . Therefore we have µ a µ b = − λ n − ( a + b ) for some integer λ . Inserting this into (15) yields2 b (1 − µ a µ b ) d = 2 a + b +1 ρ ( mod 2 n ) ⇔ b (1 − ( − λ n − ( a + b ) )) d = 2 a + b +1 ρ ( mod 2 n ) ⇔ (1 − λ n − ( a + b +1) ) d = 2 a ρ ( mod 2 n − ( b +1) ) . In case a + b = n − d = 0 a solution. Otherwise(1 − λ n − ( a + b +1) ) ∈ Z n is invertible and a solution to this equation exists. Weadditionally assume a < a for the remainder of the proof.Let ν ∈ N be an odd or zero integer such that 2 τ ν = det µ mod 2 n and write ν r := µ a a − a + 1 ∈ N + 1. Define ν l := νµ − , ν − , and note that it is either oddor zero. Then 2 τ ν l = (1 − µ a µ b ) and (15) is equal to(16) 2 b + τ ν l d = 2 a + b ν r ( mod 2 n ) ⇔ τ ν l ˆ d = 2 a ( mod 2 n − b ) , with ˆ d = ν − r d . This equation admits a solution if and only if ν l = 0 and τ ≤ a . (cid:3) Lemma 7.
Given a natural number N = 4 · j > and q ∈ k a primitive N -th rootof unity the generalised Taft algebra H q (1 , , , − has no pair in involution. Proof.
We compute the powers and coefficient matrix µ = ( µ i,j ) ≤ i,j ≤ of the pa-rameters as a = b = 0, a = b = 1 and µ , = µ , = µ , = − µ , = 1. Observethat det µ = N − · ( N −
1) mod N . By Theorem 8 a pair in involution for H q (1 , , , −
2) does not exist. (cid:3)
Remark.
Suppose q ∈ k to be a primitive p -th root of unity with p > s ∈ Z p , s = 0. The generalised Taft algebra H q (1 , s, , − s ) always has a pair ininvolution. It was shown in a previous work by Kr¨ahmer and the author, see [6],that such a pair can however only be modular if and only if s ∈ { , p − } .We have already seen that the order of the group of group-likes plays an impor-tant role for the existence of pairs in involution. Lemma 8.
Let q ∈ k be a primitive -th root of unity and H := H q ( a , a , b , b ) a generalised Taft algebra. Then H has a pair in involution.Proof. We assume that H has no pair in involution. Let a , a , b , b be the powersof the parameters and without loss of generality a ≤ a . In particular a = 1 and a + b = 1 which implies b = 1. Thus a b = 0 mod 4. Contradiction. (cid:3) A Hopf algebra is called unimodular if its distinguished group-like is equal to itscounit. Drinfeld doubles are prominent examples of such Hopf algebras.
Lemma 9.
Suppose H to be a generalised Taft algebra such that either H or H ◦ is unimodular. Then H has a pair in involution.Proof. As H has a pair in involution if and only if H ◦ has a pair in involutionwe assume without loss of generality H ◦ to be unimodular. Let 2 n · j ≥ j odd, q ∈ k be a primitive 2 n · j -th root of unity and a , a , b , b be parametersof a generalised Taft algebra, such that H ∼ = H q ( a , a , b , b ). By Lemma 5 thedistinguished group-like of H ◦ is given by g − ( a + a ) . The unimodularity of H ◦ implies a + a = 0 modulo 2 n · j . Assume that H has no pair in involution. Theequation µ , a + µ , a = 0 mod 2 n implies that either µ , = µ , = 0 or a = a . Contradiction. (cid:3) Corollary 2. If H has a quasitriangular structure it has a pair in involution.Proof. Let N ≥ q ∈ k be a primitive N -th root of unity and a , a , b , b pa-rameters of a generalised Taft algebra such that H ∼ = H q ( a , a , b , b ). Lemma6 states that N needs to be even and a = a = N . In particular a + a = 0mod N . It follows from Lemma 5 that H ◦ is unimodular. Therefore it has a pairin involution. (cid:3) Anti-Drinfeld doubles and pivotality
We investigate a Morita theoretic viewpoint on pairs in involution. This ismotivated by the comparison of Yetter-Drinfeld and anti-Yetter-Drinfeld modulesin Section 3 and the final remark of Section 2, which identified pairs in involutionwith pivotal structures on the category of Yetter-Drinfeld modules.Suppose that H and L are finite-dimensional Hopf algebras with monoidallyequivalent categories of finite-dimensional Yetter-Drinfeld modules. In this casepivotality of one such category implies pivotality of the other. Pairs in involutionmight therefore, in this strict sense, be seen as a monoidal Morita invariant property.The category of Yetter-Drinfeld modules over a finite-dimensional Hopf algebra H ENERALISED TAFT ALGEBRAS AND PAIRS IN INVOLUTION 13 can be identified with the Drinfeld center of the category of modules over H . Thisleads to the question whether Morita equivalent Hopf algebras share the propertyof having a pair in involution. Lemma 10.
Let q ∈ k be a primitive -th root of unity. The generalised Taftalgebras H := H q (34 , , , and L := H q (34 , , , are isomorphic as algebrasand L has a pair in involution whereas H does not have such.Proof. One immediately verifies that H and L are generalised Taft algebras. Let g, x, y ∈ H be the generators of H and ˆ g, ˆ x, ˆ y the generators of L . The definingrelations of the algebras H and L are g N = 1 , x |h · i| = x |h i| = 0 , y |h · i| = y |h i| = 0 ,gx = q xg, gy = q yg, xy = q · yx = q yx, ˆ g N = 1 , ˆ x |h · i| = ˆ x |h i| = 0 , ˆ y |h · i| = ˆ y |h i| = 0 , ˆ g ˆ x = q ˆ x ˆ g, ˆ g ˆ y = q ˆ y ˆ g, ˆ x ˆ y = q · ˆ y ˆ x = q ˆ y ˆ x. Therefore H and L are isomorphic as algebras.To apply Theorem 8 we compute the parameters of H and L modulo 16: a H = 34 = 2 , a H = 4 , b H = 26 = 5 · , b H = 4 a L = 34 = 2 , a L = 3 · , b L = 26 = 5 · , b L = 4 . In particular we have a H = a L = b H = b L = 1, a H = a L = b H = b L = 2. Thedeterminants and powers of the coefficient matrices are det µ H = 12 mod 16 anddet µ L = 2 mod 16 and τ H = 2 and τ L = 1 respectively. Theorem 8 implies that L has a pair in involution whereas H has none. (cid:3) The next corollary follows from the fact that the class of generalised Taft algebrasis closed under duality.
Corollary 3.
There exist generalised Taft algebras H and L such that H and L are isomorphic as coalgebras and L has a pair in involution whereas H has not. Remark.
The generalised Taft algebras H and L given in Lemma 10 provide exam-ples of Hopf algebras whose anti-Drinfeld doubles A ( H ) and A ( L ) are not Moritaequivalent in the sense of Lemma 2. That is, there are no k -linear equivalences F : A ( H )- M od → A ( L )- M od and G : A ( H )- Bimod → A ( L )- Bimod such that anatural isomorphism η : F Forg A ( H ) → Forg A ( L ) G exists. If A ( H ) and A ( L ) wereMorita equivalent in this sense A ( H ) would also be Morita equivalent (in this sense)to D ( L ). The same argument as in the proof of Lemma 2 implies that H needs tohave a pair in involution, which is a contradiction.The example given in Lemma 10 shows that studying Morita invariant propertiesof Hopf algebras in general cannot be expected to answer the question whether pairsin involution exist. However, as highlighted in the previous remark, passing to theanti-Drinfeld double and understanding its relation with the Drinfeld double seemsinteresting with regard to this question. References [1] N. Andruskiewitsch and I. Angiono. On finite dimensional Nichols algebras ofdiagonal type.
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