Group Gradings and Actions of Pointed Hopf Algebras
aa r X i v : . [ m a t h . QA ] J u l GROUP GRADINGS AND ACTIONS OF POINTED HOPFALGEBRAS
YURI BAHTURIN AND SUSAN MONTGOMERY
Abstract.
We study actions of pointed Hopf algebras on matrix algebras.Our approach is based on known facts about group gradings of matrix algebras[4] and other sources. Introduction
Many papers have been devoted to the study of actions of Hopf algebras onalgebras. However very few of them have actually classified the possible actionsof a certain Hopf algebra on a given algebra. One of the first of these paperswas the determination of the actions of the Taft algebra and its double on a onegenerator algebra [16]. More recently there has been a series of papers, such as[10, 8] on mostly semisimple Hopf algebras acting on certain rings, usually domains.Finally in [11], the actions of a Taft algebra on finite-dimensional algebras have beenstudied, in order to look at their polynomial identities.In this paper we study the actions of finite-dimensional pointed Hopf algebraswith abelian group of group-like elements on matrix algebras. Our methods arenovel, in that we use very strongly the classification of group gradings on matrices,as in [4]. We also use heavily the well-known fact (see [14] and also [15, Chapter6]) that any action of a Hopf algebra on a central simple algebra is inner. Onemotivation for this work is to try to extend the work of [19] and [7] on the HopfBrauer group to Hopf algebras which are not quasitriangular”.To state our results in more detail, we need few standard general definitions.Let H be a (finite-dimensional) Hopf algebra over a field F . Suppose that H actson a unital algebra A . This means that there is a unital left H -module structure on A written by ( h, a ) → h ∗ a such that h ∗ ε ( h )1 and h ∗ ( ab ) = ( h (1) ∗ a )( h (2) ∗ b )where h ∈ H , a, b ∈ A and ∆( h ) = h (1) ⊗ h (2) . If ◦ is another action of H on A , wesay that the actions are isomorphic if there is an algebra automorphism ϕ : A → A such that h ◦ ϕ ( a ) = ϕ ( h ∗ a ) for any h ∈ H and any a ∈ A . The actions are called equivalent if there is an automorphism of algebras ϕ : A → A and an automorphismof Hopf algebras α : H → H such that α ( h ) ◦ ϕ ( a ) = ϕ ( h ∗ a ) for any h ∈ H andany a ∈ A .We say that the action is inner if there is a convolution invertible linear map u : H → A such that, for any h ∈ H and a ∈ A we have Keywords:
Hopf algebras, graded algebras, algebras given by generators and defining relations.
Primary 16T05, Secondary 16W50, 17B37.The first author acknowledges a partial support by NSERC Discovery Grant 2019-05695. Thefirst and the second authors acknowledge support by Women in Science grant at USC. h ∗ a = X ( h ) u ( h (1) ) au − ( h (2) ) . In Section 2, we recall the classification of gradings by abelian groups on thematrix algebras. In the case where the group G of group-likes of a Hopf algebra H is semisimple, the action of G is equivalent to the grading by the dual group G .Since any finite-dimensional pointed Hopf algebra H with abelian group G of grouplikes is generated by the group like and skew-primitive elements (see the survey [2],to determine the action of H we need to know the action of skew-primitive elementson G -graded algebras. Some information about this is given in Proposition 3.1.In Section 4 we give the criterion of isomorphism of two actions of a finite-dimensional pointed Hopf algebras on a matrix algebra (Theorem 4.2) in terms oftheir respective inner actions.In Section 5 we recall the definition of a class of pointed Hopf algebras by An-ruskiewitshch - Schneider [1] in terms of generators and defining relations. Then,in Theorems 5.3, 5.5, 5.6 we determine the relations satisfied by the matrices of theinner actions of these generators. This enables us, in what follows, to approach theclassification of the actions, up to isomorphism.In Section 5.2 we determine the canonical form of these matrices in the case ofpointed Hopf algebras of rank 1, with the cyclic group of group-likes.In Section 5.3 we study the actions of a pointed Hopf algebra H on a matrixalgebra A in the case where the respective G -grading is a division grading, that is, A is a G -graded division algebra. We call such actions division actions. Theorem5.9 treats division actions in the case where H is a pointed Hopf Algebra of rank 1.In Proposition 5.10 we classify, up to isomorphism, the division actions of a pointedHopf algebra of dimension p from [1].In Section 5.11, we treat a much harder case of mixed gradings.In Section 6 we deal with the actions of the Taft algebra T n ( ω ) canonicallygenerated by g, x on the matrix algebras. Theorem 6.1 gives the classification ofsuch actions in the case where the matrix u ( x ) of the inner action of x is non-singular. In Theorem 6.2 we give a complete classification in the case of the actionsof T ( ω ) on M ( ω ).Section 7 is devoted to the study of the actions of the Drinfeld double D ( T n ( ω ))of T n ( ω ) on matrix algebras. In Section 7.1, we give the matrices for the divisionaction D ( T n ( ω )). The case of mixed action is treated, up to isomorphism, in thecase of D ( T ). In Section 7.3 we treat the case where the action of D ( T n ( ω )) iselementary.In section 8, we describe the matrix form of the actions of u q ( sl ) on the matrixalgebras. In Section 8.1 we handle the inner actions of so called Book Hopf algebrasfrom [1] and in Section 8.2 we classify the actions of u q ( sl ) on M ( F ). In Section8.3 we note that all actions of u q ( sl ) can be lifted up to the elementary actions of D ( T q )) but no division grading of D ( T q ) is a lifting of an action of u q ( sl ).2. Group gradings
Let G = G ( H ) the group of group-like elements in H . Denote by G the group ofmultiplicative F -characters of G . From now on we will be assuming that G is abelianand F is such that | G | = |G| (for instance, F algebraically closed of characteristiccoprime to |G| , or F = R and |G| = 2, etc.). In this case, it is well-known that the ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 3 algebra A acquires a G -grading(1) Γ : A = ⊕ ϕ ∈ G A ϕ given by A ϕ = { a ∈ A | g ∗ a = ϕ ( g ) a for any g ∈ G} . If a ∈ A ϕ we also write deg a = ϕ . The setSupp Γ = { ϕ ∈ G | A ϕ = { } is called the support of the grading Γ. The subspaces A γ are called homogeneouscomponents of A . An element a ∈ A γ is called homogeneous of degree γ . Onewrites deg a = γ . A subspace B is called graded if B = L γ ∈ G ( B ∩ A γ ).Group gradings on M n ( F ) have been completely described over arbitrary alge-braically closed fields F ([6], see also [17], [9, Chapter 2] and references therein).Let us call a graded algebra A graded-simple if it has no nontrivial proper gradedideals. A G -graded algebra D is called graded-division if all nonzero homogeneouselements of D are invertible. In full analogy with the classical Wedderburn - Artintheorem, the following is true. Theorem 2.1.
Let G be a group and let A be a G -graded algebra over a field F . If A is graded simple satisfying the descending chain condition on graded leftideals, then there exists a G -graded algebra D and a G -graded right D -module V such that D is a G -graded division algebra, V is finite-dimensional over D , and A is isomorphic to End D V as a G -graded algebra. Note that if { e , . . . , e m } is a graded basis of V as a right vector space over D ,with deg e i = γ i , i = 1 , . . . , n , then End D V is spanned by the elements E ij d where d is a graded element of D and E ij ( e j ) = e i . The grading is given bydeg E ij d = γ i (deg d ) γ − j . Using Kronecker products, we can write
A ∼ = M n ( F ) ⊗ D and identify E ij d with E ij ⊗ d . Then deg( E ij ⊗ d ) = γ i (deg d ) γ − j .If G is abelian then we can simply say that deg E ij d = (deg E ij )(deg d ). Weoften say that V is a right graded vector space over a graded division algebra D . If D = F , the above grading is called elementary , defined by the n -tuple γ , . . . , γ n .If dim D V = 1 then the grading is called division . In this latter case, any nonzero a ∈ A χ is invertible, for any χ ∈ G . We will use the terms elementary and division also for the actions of a Hopf algebra H , with G ( H ) abelian, on a matrix algebra M m ( F ) in the case where the respective grading of M m ( F ) by G is elementary ordivision. A simple remark is the following: Remark 2.2. If | G ( H ) | is coprime with m , then the action of H on M m ( F ) iselementary. Also if = p = char( F ) and p | m then the action of H on M m ( F ) iselementary. If the grading is elementary and dim F V = n then the grading is completelydefined by the dimension function κ : Γ → N defined as follows. Let V = ⊕ γ ∈ Γ V γ .Then κ ( γ ) = dim V γ . The group G naturally acts on the set of such functions if onedefines ( γ ∗ κ )( γ ′ ) = κ ( γγ ′ ). Two elementary gradings, defined by the dimensionfunctions κ and κ ′ , are isomorphic if and only if κ ′ ∈ G ∗ κ . Let us take a gradedbasis { e , . . . , e n } in V , with deg e i = γ i . Then κ ( γ ) is the number of elements e i BAHTURIN AND MONTGOMERY in the basis such that deg e i = γ . A graded basis of End F V is then formed by themaps E ij , as above, and deg E ij = γ i γ − j .No description of graded division algebras D over arbitrary fields is available,but in the case of an algebraically closed field of characteristic zero or coprime tothe orders of elements in G they are known to be isomorphic to the twisted groupalgebras D = F σ T , where σ ∈ Z ( T , F × ) is a 2-cocycle (actually, bicharacter) on T with values in F × . Two algebras corresponding to the cocycles σ and σ aregraded isomorphic if and only if σ and σ belong to the same cohomology classin Z ( T , F × ). In the cohomology class of a bicharacter σ there is precisely one alternating bicharacter β : T × T → F × , given by β ( ϕ, ψ ) = σ ( ϕ, ψ ) σ ( ψ, ϕ ) . An algebra D = F σ T is simple if and only if β : T × T → F × is nonsingular, that is,Ker β = { ϕ | β ( ϕ, ψ ) = 1 , for all ψ ∈ T } is trivial. In the Classification Theorem2.1, in the case of simple algebras, the graded division algebra D is simple, as anungraded algebra. As a result, D is defined, up to isomorphism, by its support T anda nonsingular alternating bicharacter β : T × T → F × . Different bicharacters leadto nonisomorphic graded division algebras D ( T , β ). Note that each homogeneouscomponent D τ of D ( T , β ) is one dimensional, and its basis { X τ } , can be chosen sothat(2) X o ( τ ) τ = I and X τ X τ = β ( τ , τ ) X τ X τ where o ( τ ) is the order of τ ∈ T . Theorem 2.3. ([4], also [9, Chapter 2])
Let T be a finite abelian group and let F be an algebraically closed field. There exists a grading on the matrix algebra M n ( F ) with support T making M n ( F ) a graded division algebra over T if and only if char F does not divide n and (3) T ∼ = Z ℓ × · · · × Z ℓ r , where ℓ · · · ℓ r = n. The isomorphism classes of such gradings are in one-to-one correspondence withnondegenerate alternating bicharacters β : T × T → F × . All such gradings belongto one equivalence class. An explicit matrix realization of D ( T , β ) is the following. First of all, using theabove equation (3) D ( T , β ) ∼ = D ( Z ℓ , β ) ⊗ D ( Z ℓ , β ) ⊗ · · · ⊗ D ( Z ℓ r , β r )where D ( Z n β ) ∼ = A = M n ( F ) and the grading on M n ( F ) is given, as follows. If Z n ∼ = ( µ ) × ( ν ), where o ( µ ) = o ( ν ) = n and ω is a primitive n th root of 1, then oneconsiders Sylvester’s clock and shift matrices C = · · · ω · · · ω · · · · · · · · · · · · · · · · · · · · · · · · ω n − , S = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · . Then the matrices X µ k ν ℓ = C k S ℓ satisfy the conditions of Equation (2), with β ( µ, nu ) = ω , and the grading of M n ( F ) is given by setting A µ k ν ℓ = Span { X µ k ν ℓ } = Span { X µ k X ν ℓ } = Span { X kµ X ℓν } , for all 0 ≤ k, ℓ < n. ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 5
In conclusion, as we noted, if A is simple then D is also a simple algebra. If thefield F is algebraically closed, D ∼ = M k ( F ). As a result, if A is a matrix algebraover an algebraically closed field, then, for some n, k , we have A ∼ = M n ( F ) ⊗ M k ( F )where the grading on M n ( F ) is elementary, while on M k ( F ) the grading is division.3. Actions on graded algebras
In this paper we are interested in the actions of finite-dimensional pointed Hopfalgebras with abelian group of group-likes on matrix algebras. We start with theactions of group algebras.3.1.
Actions of group algebras.
Here we have: H = F G is the group algebraof an abelian group G . Let A be a matrix algebra algebra and u : H → A aconvolution invertible map making this action inner. In this case, the elements u ( g ) can be explicitly written, as follows.Let, as before, G be the group of characters of G = G ( H ). According to [3] (seealso [9, Corollary 2.43], if we are given a G -grading of A = M n ( F ) ∼ = End D ( V D ),then the matrix u ( g ) of the action of for any g ∈ G , in an appropriate basis of V D ,can be written as follows. Let T ⊂ G be the support of D so that D τ = Span X τ ,for each τ ∈ T . Suppose that σ , . . . , σ s are some elements of G which are pairwisenon-congruent mod T and such that V D = ( V σ ⊗ D ) ⊕ · · · ⊕ ( V σ s ⊗ D )is the decomposition of V as a graded vector space over D , and all elements of V σ i are of degree σ i , dim V σ i = d i , i = 1 , . . . , s . Then the action of g ∈ G can be givenas the conjugation by the matrix(4) u ( g ) = ( σ ( g ) I d ⊗ X f ( g ) ) ⊕ · · · ⊕ ( σ s ( h ) I d s ⊗ X f ( g ) ) , where f ( g ) is a uniquely defined element of T such that τ ( g ) X τ = g ∗ X τ = X f ( g ) X τ X − f ( g ) = β ( f ( g ) , τ ) X τ . We have β ( f ( g ) , τ ) = τ ( g ). Since β is nonsingular, one easily checks that the map f : g f ( g ) is a well-defined homomorphism from G to T .The isomorphism classes of these actions are described by the same parametersas the isomorphism classes of corresponding gradings.3.2. Skew-primitive elements. If H is a pointed non-cosemisimple finite-dimensionalHopf algebra with an abelian group G of group-likes then there are always elements h, k ∈ G , a character χ : G → F × with χ ( g ) = 1 and an element x F G such that gxg − = χ ( g ) x for all g ∈ G , (5) ∆( x ) = x ⊗ h + k ⊗ x. (6)For an ( h, k )-primitive element x , one has ε ( x ) = 0 and S ( x ) = − k − xh − . If a Hopf algebra is generated by G and x , as above, one can replace x by a ( g, , g )-primitive element, for a suitable g ∈ G .Thus, when we study the actions of Hopf algebras, which are not group algebras,it is important to know the actions of the elements x , as above. We start with theeffect of (5). BAHTURIN AND MONTGOMERY
Proposition 3.1.
Let Γ be a grading defined in (1) . Suppose x is an element of H such that there is a character χ : G → F × satisfying hx = χ ( h ) xh , for all h ∈ G .Let K be the kernel of the action of G on A , K ⊥ its orthogonal complement in G , T the subgroup generated by Supp Γ in G . Then the following are true. (i) The action of x on A maps any A γ to A χγ . (ii) If χ
6∈ K ⊥ then x acts trivially on A . (iii) If χ T then x acts trivially on A . (iv) If there is natural m such that (Supp Γ) m = { ε } and χ m = ε then x actstrivially on A .Proof. (i) Indeed, if a ∈ A χ and h ∈ G then h ∗ ( x ∗ a ) = ( hx ) ∗ a = χ ( h )( xh ) ∗ a = χ ( h )( x ∗ ( h ∗ a ) = χ ( h ) γ ( h )( x ∗ a )= ( χγ )( h )( x ∗ a ) . It then follows that x ∗ a ∈ A χγ .(ii) Note that Supp Γ ⊂ K ⊥ . Indeed, if ϕ ∈ Supp Γ then there is 0 = a ∈ A ϕ .Take any h ∈ K then ϕ ( h ) a = ha = a . It follows that ϕ ( h ) = 1, hence ϕ ∈ K ⊥ .To prove xa = 0, for all a ∈ A , we write a ∈ A is the sum of a γ ∈ A γ , γ ∈ Supp Γ. By the above, Supp Γ ⊂ K ⊥ . If χ
6∈ K ⊥ , then there is h ∈ K such that χ ( h ) = 1. We have xa γ ∈ A χγ , by (i). Now xa g amma = h ( xa ) = ( χγ )( h )( xa ). But ( χγ )( h ) = χ ( h ) γ ( h ) = χ ( h ) = 1. Thus xa = 0,as claimed.(iii) Follows easily from (ii), considering K ⊥ is a subgroup of G .(iv) In this case, also T m = { } so that (iii) applies. (cid:3) Taft pairs and algebras.
Now let H contain a group-like element g and a(1 , g )-primitive element x such that(7) g n = 1 , x n = 0 , gx = ωxg, where ω is a primitive n th root of 1 . We have ∆( g ) = g ⊗ g, ∆( x ) = x ⊗ g ⊗ x,ε ( g ) = 1 , ε ( x ) = 0 , S ( g ) = g − , S ( x ) = − g − x. One calls such a pair of elements a
Taft ω -pair . In this case, g s x = χ ( g s ) g s x , where χ ( g s ) = ω s . Let G be the cyclic subgroup generated by g .An n -dimensional Hopf algebra generated by an ω -pair g, x is called an n -dimensional Taft ω -algebra , denoted by T n ( ω ), or simply T n , if ω is fixed.Since in this case G ∼ = Z n , it follows from the above, that A acquires a Z n -grading, as follows. Let χ ∈ G be given by χ ( g ) = ω . Set A k = A χ k . Then A = ⊕ n − k =0 A k is the desired grading. Corollary 3.2.
If the action of G is not faithful then the action of x is trivial. Ifthe support of the grading is contained in a proper subgroup T of G then the actionof x is trivial.Proof. Let e = g k , k | n act trivially on A . Then the support of the grading iscontained in proper subgroup T = ( g k ) ⊥ of G , whereas χ generates G . So we canuse Proposition 3.1 (ii). (cid:3) ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 7 Inner actions on central algebras
An algebra A with identity element 1 is called central if its center coincides with F .
1. Suppose that a Hopf algebra H acts on A , and this action is inner via theconvolution invertible map u : H → A . Then, for any group-like element g ∈ G andany ( h, k )-primitive element x , h, k ∈ G , one has(8) g ∗ a = u ( g ) au ( g ) − , x ∗ a = u ( x ) au ( h ) − − u ( k ) au ( k ) − u ( x ) u ( h ) − . In the case where x is (1 , k )-primitive, we would have(9) g ∗ a = u ( g ) au ( g ) − , x ∗ a = u ( x ) a − u ( k ) au ( k ) − u ( x ) . In the case where x is ( h, g ∗ a = u ( g ) au ( g ) − , x ∗ a = u ( x ) au ( h ) − − au ( x ) u ( h ) − = [ u ( x ) , a ] u ( h ) − . Here [ a, b ] = ab − ba is the usual commutator of a, b ∈ A . Lemma 4.1. If g, x act as in (9) , then setting u ′ ( x ) = u ( x ) − λu ( g ) , u ′ ( g ) = µu ( g ) does not change the action. In the case of (10) , the action is preserved if we replace u ( x ) by u ( x ) − λ.I and u ( g ) by µu ( g ) . In both cases, λ ∈ F and µ ∈ F \ { } ,Proof. Indeed, if we denote the “old” action by ∗ and the “new” one by ◦ , then g ◦ a = u ′ ( g ) au ′ ( g ) − = u ( g ) au ( g ) − = g ∗ ax ◦ a = u ′ ( x ) a − u ′ ( g ) au ′ ( g ) − u ′ ( x )= ( u ( x ) − λu ( g )) a − u ( g ) au ( g ) − ( u ( x ) − λu ( g ))= u ( x ) a − u ( g ) au ( g ) − u ( x ) = x ∗ a, as claimed.The case of (10) is even simpler. (cid:3) A much more general isomorphism result about the actions of pointed Hopfalgebras is the following.
Theorem 4.2.
Let H be a finite-dimensional pointed Hopf algebra with abeliangroup of group-likes, A a matrix algebra. If two inner actions ∗ and ◦ defined bythe functions u : H → A and v : H → A are isomorphic then there exists aninvertible element C ∈ A and λ ( g ) , µ ( g, x ) ∈ F , such that u ( g ) = λ ( g ) Cv ( g ) C − ,where λ ( g ) ∈ F , for any grouplike g ∈ H and u ( x ) = Cv ( x ) C − + µ ( g, x ) u ( g ) , µ ( g, x ) ∈ F , for any (1 , g ) -primitive element x ∈ H . In the case of a ( g, -primitiveelement y , we must have u ( y ) = Cv ( y ) C − + µ ( g, y ) I .Proof. If g ∈ G ( H ), then ϕ : A → A is an isomorphism of actions, then g ∗ ϕ ( A ) = ϕ ( g ◦ A ), for any A ∈ A . Since A is central simple, there is invertible C ∈ A suchthat ϕ ( A ) = CAC − . So we have u ( g ) CAC − u ( g ) − = Cv ( g ) Av ( g ) − C − , forall A ∈ A . As a result, v ( g ) − C − u ( g ) C − is a central element of A . Since A iscentral, v ( g ) − C − u ( g ) C − = λ ( g ) ∈ F and u ( g ) = λ ( g ) Cv ( g ) C − , as claimed. Wecan also write u ( g ) C = λ ( g ) Cv ( g ) and v ( g ) − C − = λ ( g ) C − u ( g ) − .Now let x be a (1 , g )-primitive element of H and suppose x ∗ ϕ ( A ) = ϕ ( x ◦ A ).Then u ( x ) CAC − − u ( g ) CAC − ( g ) − u ( x ) = C ( v ( x ) A − v ( g ) Av ( g ) − v ( x )) C − . BAHTURIN AND MONTGOMERY
It follows that u ( x ) CAC − − Cv ( x ) AC − = u ( g ) CAuC − ( g ) − u ( x ) − v ( g ) Av ( g ) − v ( x )) C − ( u ( x ) C − Cv ( x )) A = Cv ( g ) A ( λC − u ( g ) − u ( x ) C − v ( g ) − v ( x ))( v ( g ) − C − u ( x ) C − v ( g ) − v ( x )) A = A ( λC − u ( g ) − u ( x ) − v ( g ) − v ( x ))( v ( g ) − C − u ( x ) C − v ( g ) − v ( x )) A = A ( v ( g ) − C − u ( x ) C − v ( g ) − v ( x )) . As a result, v ( g ) − C − u ( x ) C − v ( g ) − v ( x ) is a central element in A , and so thereexists µ ( g, x ) ∈ C such that u ( x ) = Cv ( x ) C − + µ ( g ) u ( g ), as claimed. (cid:3) Conversely, suppose our conditions hold and H is generated by its group-likesand skew-primitives. Then v ( g ) − C − u ( g ) C − and v ( g ) − C − u ( x ) C − v ( g ) − v ( x )are central elements. We can read our calculations in the reverse order, to find that ϕ ( A ) = CAC − preserves the action of group-like and skew-primitive elements.Since H is generated by these elements, we find that ϕ is an isomorphism of actions ∗ and ◦ . 5. Pointed Hopf algebras
Pointed Hopf algebras with abelian coradical have been classified thanks to ef-forts of many authors, specially N. Andruskiewitchsh, H.-J. Schneider, I. Hecken-berger and I. Angiono. A useful survey of the development in the area is [2]. Inthis paper we will be looking at actions of a class oh pointed Hopf algebras in thepaper [1].Let G be a finite nontrivial abelian group. A datum for a quantum linear space R = R ( a , . . . , a n ; χ , . . . , χ n ) consists of the elements a , . . . , a n ∈ G , χ , . . . , χ n ∈ G = b G , such that the following conditions are satisfied: q j = χ j ( a j ) = 1(11) χ j ( a i ) χ i ( a j ) = 1 for all 1 ≤ i = j ≤ n. (12)One says that the datum, or its associated quantum linear space, has rank n . If N j = o ( q j ), the order of q i in G , then R = R ( a , . . . , a n ; χ , . . . , χ n ) is an algebrawith the following presentation in terms of generators and defining relations:(13) R = h x , . . . , x n | x N i i = 0; x i x j = χ j ( a i ) x j x i for all 1 ≤ i = j ≤ n i . Let us fix a decomposition G = ( g ) × · · · × ( g k ) and denote by M ℓ the order of g ℓ , 1 ≤ ℓ ≤ k . Let a , . . . , a n ∈ G , χ , . . . , χ n ∈ G be the datum for a quantumlinear space R = R ( a , . . . , a n ; χ , . . . , χ n ), that is, (11, 12) hold.A compatible datum D for G and R consists of a vector ( µ , . . . , µ n ), µ i ∈ { , } and a matrix ( χ ij ), λ ij ∈ F , 1 ≤ i, j ≤ n , such that(i) if a N i i = 1 or χ N i i = ε , then µ i = 0,(ii) if a i a j = 1 or χ i χ j = ε , then λ ij = 0. Definition 5.1.
Let G be a finite abelian group, R a quantum linear space, and D a compatible datum. Keep the notation above. Let P ( G , R , D ) be the algebra ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 9 presented by generators g ℓ , ≤ ℓ ≤ k , and x i , ≤ i ≤ n , with defining relations g M ℓ ℓ = 1 , ≤ ℓ ≤ k ;(14) g ℓ g t = g t g ℓ , ≤ ℓ, t ≤ k ;(15) x i g ℓ = χ i ( g ℓ ) g ℓ x i , , ≤ ℓ ≤ k ;(16) x N i i = µ i (1 − a N i i ) , ≤ i ≤ n ;(17) x j x i = χ i ( a j ) x i x j + λ ij (1 − a i a j ) , ≤ i, j ≤ n. (18) A unique Hopf algebra structure on P ( G , R , D ) is given by ∆( g ℓ ) = g ℓ ⊗ g ℓ , ∆( x i ) = x i ⊗ a i ⊗ x i , ≤ ℓ ≤ k, ≤ i, j ≤ n ; S ( g ℓ ) = g − ℓ , S ( x i ) = − a − i x i , ≤ ℓ ≤ k, ≤ i ≤ n. One can also say that H is generated by its group G = G ( H ) of group-likes,which is abelian, and the set of (1 , a i )-primitive x i , for some element a i ∈ G , i = 1 , . . . , n . These elements satisfy the above relations (16, 17, 18). In theserelations, χ , . . . , χ n are multiplicative characters χ i : G → F × , N i = o ( χ i ( a i )) = 1.If one denotes q i = χ i ( a i ), then N i is the order of q i in the group F × . The elements µ i and λ ij are subject to the compatibility conditions (i), (ii), just before Definition5.1.Finally, the number n of generators of the quantum vector space R is called the rank of the Hopf algebra P ( G , R , D ). Remark 5.2.
Let P ( G , R , D ) act on an algebra A so that, for some i = 1 , . . . , n ,the action of x i is nonzero. If g acts trivially on A then χ i ( g ) = 1 . If a i actstrivially then x i acts trivially, too. This follows by Proposition 3.1 and by (11).5.1.
Inner actions of pointed Hopf algebras P ( G , R , D ) . In this section westudy the actions of pointed Hopf algebras of rank one on matrix algebras. We set H = P ( G , R , D ), as above. Theorem 5.3.
Let u : H → A define an inner action of a Hopf algebra H on amatrix algebra A . Then for each i = 1 , . . . , k , there is a function λ i : G → F suchthat, for any g ∈ G one has (19) u ( g ) u ( x i ) u ( g ) − = χ i ( g ) u ( x i ) + λ i ( g ) u ( a i ) . One can choose u so that λ ( g ) = 0 , that is, (20) u ( g ) u ( x i ) = χ i ( g ) u ( x i ) u ( g ) , for all g ∈ G , the action remaining the same.Proof. Note that since G is abelian, we have a i ∗ ( g ∗ A ) = g ∗ ( a i ∗ A ), for any g ∈ G , A ∈ A . Thus u ( g ) u ( a i ) Xu ( a i ) − u ( g ) − = u ( a i ) u ( g ) Xu ( g ) − u ( a i ) − . Now let us compute g ∗ ( x i ∗ A ) = χ i ( g ) x i ∗ ( g ∗ A ) in terms of the inner action viathe map u : H → A . We have g ∗ ( x i ∗ A ) = u ( g )( u ( x i ) A − u ( a i ) Au ( a i ) − u ( x i )) u ( g ) − = u ( g ) u ( x i ) Au ( g ) − − u ( g ) u ( a i ) Au ( a i ) − u ( x i ) u ( g ) − = u ( g ) u ( x i ) Au ( g ) − − u ( a i ) u ( g ) Au ( g ) − u ( a i ) − u ( g ) u ( x i ) u ( g ) − (21) χ i ( g ) x i ∗ ( g ∗ A ) = χ i ( g ) x i ∗ ( u ( g ) Au ( g ) − )= χ i ( g ) u ( x i ) u ( g ) Au ( g ) − − χ i ( g ) u ( a i ) u ( g ) Au ( g ) − u ( a ) − u ( x i )(22)If we subtract (22) from (21), we obtain0 = ( u ( g ) u ( x i ) − χ i ( g ) u ( x i ) u ( g )) Au ( g ) − − u ( a i ) u ( g ) A ( u ( g ) − u ( a i ) − u ( g ) u ( x i ) − χ ( g ) u ( g ) − u ( a i ) − u ( x i ) u ( g )) u ( g ) − . Multiplying on the left by u ( g ) − u ( a i ) − and on the right by u ( g ), we obtain( u ( g ) − u ( a i ) − u ( g ) u ( x i ) − χ i ( g ) u ( g ) − u ( a i ) − u ( x i ) u ( g )) A = A ( u ( g ) − u ( a i ) − u ( g ) u ( x i ) − χ ( g ) u ( g ) − u ( a i ) − u ( x i ) u ( g )) . Since A is arbitrary and A central simple there is λ i ( g ) ∈ F such that u ( g ) − u ( a i ) − u ( g ) u ( x i ) = χ ( g ) u ( g ) − u ( a i ) − u ( x i ) u ( g )) + λ i ( g )1 A ,u ( g ) u ( x i ) u ( g ) − = χ i ( g ) u ( x i ) + λ i ( g ) u ( a i ) , proving (19).If g , g ∈ G , then u ( g ) u ( x i ) − χ i ( g ) u ( x i ) u ( g ) = λ i ( g ) u ( a i ) u ( g )(23) u ( g ) u ( g ) u ( x i ) − χ i ( g ) u ( g ) u ( x i ) u ( g ) = λ i ( g ) u ( g ) u ( a i ) u ( g )Since u ( g g ) is a scalar multiple of u ( g ) u ( g ), we can write u ( g g ) u ( x i ) − χ i ( g g ) u ( x i ) u ( g g ) = λ i ( g g ) u ( a i ) u ( g g ) u ( g ) u ( g ) u ( x i ) − χ i ( g g ) u ( x i ) u ( g ) u ( g ) = λ i ( g ) u ( a i ) u ( g ) u ( g )It follows that χ i ( g ) u ( g ) u ( x i ) u ( g ) + λ i ( g ) u ( g ) u ( a i ) u ( g )= χ i ( g ) χ i ( g ) u ( x i ) u ( g ) u ( g ) + λ i ( g g ) u ( a i ) u ( g ) u ( g )(24)Cancelling out u ( g ) and substituting (23), where g is replaced by g , to (24), weobtain χ i ( g ) χ i ( g ) u ( x i ) u ( g ) + χ i ( g ) λ i ( g ) u ( a ) u ( g ) + λ i ( g ) u ( g ) u ( a i )= χ i ( g ) χ i ( g ) u ( x i ) u ( g ) + λ i ( g g ) u ( a ) u ( g ) . It follows that χ i ( g ) λ i ( g ) u ( a i ) u ( g ) + λ i ( g ) u ( g ) u ( a i ) = λ i ( g g ) u ( a i ) u ( g ) . If we set u ( g ) u ( a i ) = ξ ( g, a i ) u ( a i ) u ( g ), for ξ ( g, a i ) ∈ F , then we get λ i ( g g ) = λ i ( g ) χ i ( g ) + λ i ( g ) ξ ( g , a i ) λ ( g g ) = λ ( g ) χ i ( g ) + λ i ( g ) ξ ( g , a i ) λ ( g )( ξ ( g , a i ) − χ i ( g )) = λ ( g )( ξ ( g , a i ) − χ ( g ))If g = g , g = a i then(25) λ i ( g )(1 − χ i ( a i )) = λ ( a i )( ξ ( g, a i ) − χ i ( g )) . Since χ i ( a i ) = 1, it follows that once λ i ( a ) = 0, all λ i ( g ) = 0. Now if we replace u ( x i ) by ¯ u ( x i ) = u ( x i ) + λ ( a i )1 − χ i ( a i ) , then we will have u ( a i )¯ u ( x i ) = χ i ( a i )¯ u ( x i ) u ( a i ).Our calculation shows that then also u ( g )¯ u ( x i ) − χ i ( g )¯ u ( x i ) u ( g ), for all g ∈ G , asclaimed. (cid:3) ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 11
An important consequence of this theorem is the following.
Corollary 5.4.
Let a Hopf algebra H with a group of group-likes G and G = b G acton a matrix algebra A . Let Γ : A = M ϕ ∈ G A ϕ be the associated G grading of A . If x is a (1 , a ) -primitive element in H for some a ∈ G and χ ∈ G is such that gx = χ ( g ) x , for all g ∈ G . Then u ( x ) ∈ A χ . Let H = P ( G , R , D ), as above. Theorem 5.5.
Let u : H → A define an inner action of a Hopf algebra H on amatrix algebra A . Then there is σ i ∈ F such that the following relation holds forthe elements u ( a i ) and u ( x i ) of the algebra A : (26) u ( x i ) N i = µ i . A + σ i u ( a i ) N i . Proof.
This is an application of the q -binomial formula [18, Lemma 3]. Let L y , R z : A → A denote the left, respectively, right multiplications by y , respectively z , inthe algebra A . Then we can rewrite the action of x on any A ∈ A in Equation (9)as x i ∗ A = ( L u ( x i ) − L u ( a i ) R u ( a i ) − R u ( x i ) )( A )hence x N i i ∗ A = ( L u ( x ) − L u ( a ) R u ( g ) − R u ( x ) ) N i ( A ) . Setting ϕ = L u ( x ) , ψ = − L u ( a ) R u ( a ) − R u ( x ) , we get ψϕ = − L u ( a i ) R u ( a i ) − R u ( x i ) L u ( x i ) = − L u ( a i ) L u ( x i ) R u ( a i ) − R u ( x i ) = − L u ( a i ) u ( x i ) R u ( a i ) − R u ( x i ) = − q − i L u ( x i ) u ( a i ) R u ( a i ) − R u ( x i ) = − q − i L u ( x i ) L u ( a i ) R u ( a i ) − R u ( x i ) = − q − i ϕψ. Applying the q -binomial formula to ( ϕ + ψ ) n , we get x N i i ∗ A = ( L u ( x i ) Ni + ( − N i ( L u ( a i ) R u ( a i ) − R u ( x i ) ) N i ) A = u ( x i ) N i A + ( − N i u ( a i ) N i R N i u ( a i ) − u ( x i ) A = u ( x i ) N i A + ( − N i R ( u ( a i ) − u ( x i )) Ni A. Now ( u ( a i ) − u ( x i )) N i = q Ni ( Ni − i u ( x i ) N i u ( a i ) − N i So, taking into account that ( − N i q Ni ( Ni − i = − , for any natural N i , we get µ i (( A − u ( a i ) N i Au ( a i ) − N i ) = µ i (1 − a N i i ) ∗ A = u ( x i ) N i A − u ( a i ) N i Au ( a i ) − N i u ( x ) N i . This can be rewritten as Au ( a i ) − N i ( − µ i . A + u ( x i ) N i ) = u ( a i ) − N i ( − µ i . A + u ( x i ) N i ) A Since A is central simple and A ∈ A is arbitrary, there is σ i ∈ F such that u ( a i ) − N i ( − µ i . A + u ( x i ) N i ) = σ i . A . Multiplying both sides by u ( a i ) N i on the left, we obtain the desired equation (26). (cid:3) We keep setting H = P ( G , R , D ), as before. Theorem 5.6.
The following relations hold for the elements of the inner action of H = P ( G , R , D ) on a matrix algebra A , defined by the convolution invertible map u : H → A . For any ≤ i, j ≤ n there exist ζ ij ∈ F such that (27) u ( x j ) u ( x i ) − χ j ( a i ) u ( x i ) u ( x j ) = λ ij . A + ζ ij u ( a i ) u ( a j ) . Proof.
For the proof, we are going to use the defining relation (18). We comparethe action of x j x i − χ i ( a j ) x i x j with the action of λ ij (1 − a i a j ) on the elements of A . Let G be the group of group-likes of H , G the dual group for G . Then A isgraded by G . Thus, it is sufficient to check (27) on the homogeneous elements ofthe grading. Let A ∈ A δ , δ ∈ G . Then x i ∗ A ∈ A χ i δ . So x j ∗ ( x i ∗ A ) = u ( x j )( x i ∗ A ) − ( χ i δ )( a j )( x i ∗ A ) u ( x j )= u ( x j ) [ u ( x i ) A − δ ( a i ) Au ( x i )] − ( χ j δ )( a j ) [ u ( x i ) A − δ ( a i ) Au ( x i )] u ( x j )= u ( x j ) u ( x i ) A − δ ( a i ) u ( x j ) Au ( x i ) − ( χ i δ )( a j ) u ( x i ) Au ( x j ) + δ ( a j ) δ ( a i ) Au ( x i ) u ( x j ) . (28)Likewise, x i ∗ ( x j ∗ A ) = u ( x i )( x j ∗ A ) − ( χ j δ )( a i )( x j ∗ A ) u ( x i )= u ( x i ) [ u ( x j ) A − δ ( a j ) Au ( x j )] − ( χ j δ )( a i ) [ u ( x j ) A − δ ( a j ) Au ( x j )] u ( x i )= u ( x i ) u ( x j ) A − δ ( a j ) u ( x i ) Au ( x j ) − ( χ j δ )( a i ) u ( x j ) Au ( x i ) + δ ( a i ) δ ( a j ) Au ( x j ) u ( x i ) . (29)We then have χ i ( a j )( x i ∗ ( x j ∗ A )) = χ i ( a j ) u ( x i ) u ( x j ) A − χ i ( a j ) δ ( a j ) u ( x i ) Au ( x j ) − χ i ( a j )( χ j δ )( a i ) u ( x j ) Au ( x i )+ χ i ( a j ) δ ( a i ) δ ( a j ) Au ( x j ) u ( x i ) . Note χ i ( a j ) χ j ( a i ) = 1. As a result, x j ∗ ( x i ∗ A ) − χ i ( a j )( x i ∗ ( x j ∗ A ))= [ u ( x j ) u ( x i ) − χ i ( a j ) u ( x i ) u ( x j )] A +[ δ ( a i a j ) Au ( x j ) u ( x i ) − δ ( a i a j ) χ i ( a j ) Au ( x i ) u ( x j )] . (30)Now let us compute the action of λ ij (1 − a i a j ) on A ∈ A δ . We have λ ij (1 − a i a j ) ∗ A = λ ij (1 − δ ( a i a j )) A = λ ij A − λ ij u ( a i ) u ( a j ) Au ( a j ) − ( a i ) − . (31)When we equate (30) and (31) and rearrange terms, we will have ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 13 [ u ( x j ) u ( x i ) − χ i ( a j ) u ( x i ) u ( x j ) − λ ij ] A = δ ( a i a j ) A [ u ( x j ) u ( x j ) − χ i ( a j ) u ( x i ) u ( x j ) − λ ij ]= u ( a i ) u ( g j ) Au ( a j ) − u ( a i ) − [ u ( x j ) u ( x i ) − χ i ( a j ) u ( x i ) u ( x j ) − λ ij ]or u ( a j ) − u ( a i ) − [ u ( x j ) u ( x i ) − χ i ( a j ) u ( x i ) u ( x j ) − λ ij ] A = Au ( a j ) − u ( a i ) − [ u ( x j ) u ( x i ) − χ i ( a j ) u ( x i ) u ( x j ) − λ ij ]Thus the element u ( a j ) − u ( a i ) − [ u ( x j ) u ( x i ) − χ i ( a j ) u ( x i ) u ( x j ) − λ ij ]is central in A , so that there exists ζ ij ∈ F such that u ( a j ) − u ( a i ) − [ u ( x j ) u ( x i ) − χ i ( a j ) u ( x i ) u ( x j ) − λ ij ] = ζ ij . A , or u ( x j ) u ( x i ) − χ i ( a j ) u ( x i ) u ( x j ) = λ ij . A + ζ ij u ( a i ) u ( a j ) . (cid:3) Theorems 5.3, 5.5 and 5.6 work for Taft algebras and their doubles, but alsoLusztig’s kernel and so on. Not only do they give the necessary but also they givesufficient conditions for the actions of H on matrix algebras. Although it seemsimpossible to classify all sets of matrices satisfying all the relations given in thesetheorems, some interesting examples can still be provided. This is done in theremainder of the paper.5.2. Matrix form for the actions of pointed Hopf algebras of rank 1, withcyclic group of group-likes.
Let G be a cyclic group generated by an element g of order m , with the group of characters G . Then there is γ ∈ G such that γ ( g ) = ω ,a primitive m th root of 1. Let R be a quantum torus of rank 1, with the compatibledatum D = a, χ, x , where a ∈ G , χ ∈ G such that q = χ ( a ) = 1, n = o ( q ). In theHopf algebra H = P ( G , R , D ) we have generators g, x such that gx = χ ( g ) xg and x n = µ (1 − a n ). If a n = 1 or χ n = ε , we must have µ = 0. Otherwise, any Hopfalgebra is isomorphic to the one with either µ = 0 or µ = 1. The algebras with µ = 0 are sometimes called of nilpotent type.Let us consider the action of such H on a matrix algebra A . Then A = End V ,where V is a finite-dimensional G -graded vector space over F : V = L ϕ ∈ G V ϕ . If f ∈ A τ then f ∗ A ϕ ⊂ A τϕ . As a result, x ∗ A ϕ ⊂ A χϕ . We can always choose u ( g )so that u ( g ) m = I . It was shown in Theorem 5.3 that u ( x ) can be chosen so that u ( g ) u ( x ) u ( g ) − = χ ( g ) u ( x ). So, u ( x ) ∈ A χ .There is a primitive m th root of 1 ζ ∈ F and a vector space decomposition V = P m − s =0 V s such that u ( g ) | V s = ζ s id V s . Let ℓ be the order of χ and m = ℓr . Werearrange the subspaces V s so that(32) V = r − X t =0 W t where W t = ℓ − X k =0 V t + kr . Then each W t is invariant with respect to u ( x ), which follows from u ( g ) u ( x ) = ζ r u ( x ) u ( g ). In an appropriate basis, the restriction u ( x ) ( t ) of u ( x ) to each W t hasthe form(33) u ( x ) ( t ) = . . . u ( t )1 ℓ u ( t )21 . . . . . . . . . . . . . . . . . . . . . u ( t ) ℓ . where each u ( t ) i +1 ,i is a rectangular d ( t ) k +1 × d ( t ) k matrix of rank r ( t ) i . Here dim V t + kr = d ( t ) k and r ( t ) i = rank u ( t ) i +1 ,i .Further restrictions on u ( x ) appear when one uses Theorem 5.5 and compares u ( x ) n with µ. id V + λu ( g ) n .First assume that one of the conditions a n = 1 or χ n = ε holds. Then H isnilpotent. We have u ( x ) n = λu ( a ) n . Now n is a divisor of ℓ and if n = ℓ (whichis the same as χ n = ε ), we must have simply u ( x ) n = 0, because u ( a ) n is diagonaland u ( x ) n does not have nonzero diagonal blocks. Also, if χ n = ε (hence ℓ = n )and a n = 1 then u ( a ) n is a scalar matrix and hence u ( x ) n is a scalar matrix. So inthis case, u ( x ) n = λ.I is a scalar matrix. It is hard to say much if λ = 0. But if λ = 0 then u ( x ) is a nonsingular map.One easily concludes the all the maps u , u , . . . , u ℓ , are isomorphisms. Indeed,the entries of ( u ( x ) ( t ) ) n are all cyclic permutations of u u · · · u ℓ . Since ( u ( x ) ( t ) ) n is a nonzero scalar matrix, we have rank ( u ( x ) ( t ) ) n = d ( t )1 + · · · d ( t ) ℓ . If d ( t ) k is thesmallest of the sizes of these blocks, then the rank of each diagonal entry is at mostthis number. It follows that all V t + kr have the same dimension and all the blocksare nonsingular. By an appropriate choice of the bases in these subspaces we mayassume that u ( x ) ( t ) has the form(34) u ( x ) ( t ) = . . . λI d t I d t . . . . . . . . . . . . . . . . . . . . . I d t . If we assume that both a n = 1 and χ n = ε , then we either have x n = 0 or x n = 1 − a n . In the first case, u ( x ) n = λu ( a ) n , in the second u ( x ) n = I + λu ( a ) n ,for some λ ∈ F .In both cases, on each W t , the action of µ. id V + λu ( a ) n is scalar with the coeffi-cient µ t = µ + λζ tn . At the same time, ( u ( x ) ( t ) ) n is a block-diagonal matrix, withblocks of the same size as those for u ( a ). If µ t = 0, then ( u ( x ) ( t ) ) n = 0 and wecannot say much more. Otherwise, the same argument as the one leading to (34)works and we have that(35) u ( x ) ( t ) = . . . µ t I d t I d t . . . . . . . . . . . . . . . . . . . . . I d t . The results obtained above can be summarized as follows.
ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 15
Theorem 5.7.
Let a Hopf algebra H = P ( G , R , D ) with G = ( g ) , R = F [ x ] , bedefined by g m = 1 , gx = χ ( g ) xg and x n = µ (1 − a n ) . Suppose H acts on a matrixalgebra A = End V . Then the action is inner, via u : H → A . There is an m thprimitive root ζ of such that V = P m − i =0 V i , where V i is an eigenspace for u ( a ) ,with eigenvalue ζ i . If m = nr then V = P rt =0 W t , each W t being invariant underboth u ( g ) and u ( x ) . In an appropriate basis of W t , the matrix of u ( x ) has the formof (33), while the matrix of u ( a ) is scalar with coefficient ζ t + kr . If H is nilpotent,then either u ( x ) n = 0 or u ( x ) takes the form (34). If H is not nilpotent and u ( x ) acts on W t in a nonsingular way, then, the matrix u ( x ) ( t ) of this restriction takesthe form of (35). Remark 5.8.
Note that, up to some change of generators, which we discuss at thebeginning of Section 7, this applies to all, but one, types of pointed Hopf algebrasof dimension p from the classification in [1]. The exceptional case (a) from thatpaper is discussed in Section 5.3.1. The actions in the case of the “popular” Taftalgebras [11] are discussed in detail in Section 6.5.3. Actions of H and division gradings. Let H act on a matrix algebra A , G = G ( H ), χ ∈ G = b G , a ∈ G , x a (1 , a )-primitive element satisfying gx = χ ( g ) x and q = χ ( a ) is of order n = 1. Suppose the action of G on A makes A a gradeddivision algebra. Let T ⊂ G be the support of the grading. Since the grading isdivision, T is a subgroup, whose order is a square m , where m , hence m are thedivisors of the order of G . The grading is accompanied by an alternating bicharacter β : T × T → F × . The basis of A is formed by the elements X ϕ , ϕ ∈ T , X o ( ϕ ) ϕ = I ,and X ϕ X ψ = β ( ϕ, ψ ) X ψ X ϕ , for all ϕ, ψ ∈ T . Also, o ( ϕ ) is the order of ϕ in thegroup G .As mentioned earlier, the action of g ∈ G is conjugation by a matrix u ( g ) = X f ( g ) such that β ( f ( g ) , ϕ ) = ϕ ( g ), for any ϕ ∈ T . From Proposition 5.3 it follows that u ( g ) u ( x ) u ( g ) − = χ ( g ) u ( x ) + λ ( g ) u ( a ). We know that u ′ ( x ) = u ( x ) − λ − χ ( a ) u ( a )satisfies u ( g ) u ′ ( x ) u ( g ) − = χ ( g ) u ′ ( x ). If we write u ′ ( x ) = P ϕ ∈ T α ϕ X ϕ then X ϕ ∈ T α ϕ X f ( g ) X ϕ = X ϕ ∈ T χ ( g ) α ϕ X ϕ X f ( g ) = X ϕ ∈ T α ϕ β ( f ( g ) , ϕ ) X ϕ X f ( g ) = X ϕ ∈ T χ ( g ) α ϕ X ϕ X f ( g ) . Comparing like terms on both sides of the above equation, and considering β ( f ( g ) , ϕ ) = ϕ ( g ), we find α ϕ ϕ ( g ) = χ ( g ) α ϕ . Thus α ϕ = 0 implies ϕ ( g ) = χ ( g ),for all g . Hence u ′ ( x ) = αX χ . As a result,(36) u ( x ) = αX χ + λ − χ ( a ) X f ( a ) . The value of the coefficient of X f ( a ) is unimportant, so we simply write u ( x ) = αX χ + γX f ( a ) . Now we need to check that x n and µ (1 − a n ) act on A in the same way. The necessaryand sufficient conditions for this are given in Proposition 5.5: µ · I + u ( x ) n = σu ( a ) n .Note that if a n = e or χ n = ε then µ = 0. Otherwise, µ can be arbitrary, but actually, one can only consider the cases µ = 0 or µ = 1. Let us start with the case µ = 0. In this case, one should have( αX χ + γX f ( a ) ) n = σX nf ( a ) Using the q -binomial formula, we get α n X nχ + γ n X nf ( a ) = σ n X nf ( a ) . This shows that there was no need to view u ( x ) in its general form αX χ + γX f ( a ) rather than simply u ( x ) = αX χ . So we can continue as follows α n X nχ = σ n X nf ( a ) . So χ n = f ( a n ) is a necessary condition in this case. Since σ is arbitrary, thiscondition is also sufficient.Now let µ = 1, so that a n = e and χ n = ε . In this case, we have I + α n X χ n = σ n X nf ( a ) . Hence, (1 + α n ) I = σ n X nf ( a ) . So we see that this can happen only when f ( a n ) = ε , which is the same as a n acts trivially on A . It follows that if such a Hopf algebra acts on a graded divisionalgebra, this action cannot be faithful, even if we consider only group-like elements. Theorem 5.9.
Let H = P ( G , R , D ) be a pointed Hopf algebra of rank 1, withadmissible data D = ( a, χ, µ , o ( χ ( a )) = n . Let G be the group of characters on G .Assume that the action of G on a matrix algebra A makes A a G -graded divisionalgebra, with support T . In any extension of this action of G to H we must have u ( x ) = αX χ , for some α ∈ F . If χ T then the action of x is zero. If χ ∈ T and χ n = f ( a ) n then extensions always exist, with any α ∈ F . If µ = 0 , this conditionis also necessary. If µ = 1 , in which case χ n = ε and a n = e , the extensions existeven if a n acts non-trivially. In this case, α n = − .The actions via u ( x ) = αX χ and u ′ ( x ) = α ′ X χ are isomorphic if and only if α = α ′ . An example of a p -algebra. In this section we consider the division actionsof an algebra, which is Example (a) in the classification of pointed Hopf algebrasof dimension p in [1]. Given an odd prime number p >
2, this is an algebra ofthe form T p ( ω ) ⊗ FZ p . In other words, this is a pointed Hopf Algebra H , of thetype described in Section 5, where G = ( g ) × ( h ) is an elementary abelian p -group, χ : G → F × given by χ ( g ) = ω , χ ( h ) = 1, µ = 0. Let us consider the action of H on a matrix algebra A where the action of G = G ( H ) induces a grading by G ,which is a division grading. We have G = ( µ ) × ( ν ), where µ ( g ) = ω , µ ( h ) = 1, ν ( g ) = 1, ν ( h ) = ω . The grading is completely defined by an alternating bicharacter β : G × G → F × , given by β ( µ, ν ) = τ where τ is a p th primitive root of 1. In thiscase, A = M p ( F ), is generated by the clock and shift matrices X µ and X ν and, asmentioned earlier, we may assume ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 17 X µ = diag(1 , τ, τ , . . . , τ p − ) X ν = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ;These matrices satisfy X pµ = X pν = I and X ν X µ = β ( µ, ν ) X µ X ν .Using Proposition 5.3, we can say that the degree of u ( x ) is χ , and so there isscalar α ∈ F such that u ( x ) = αX χ . Now χ = µ and so u ( x ) = αX µ . As for u ( g )and u ( h ), these can be found when we consider that the actions of g and h areconjugation by the matrices X f ( g ) and X f ( h , given by x ∗ X ϕ = X f ( g ) X ϕ X − f ( g ) = β ( ϕ, f ( g )) X ϕ ,y ∗ X ϕ = X f ( h ) X ϕ X − f ( h ) = β ( ϕ, f ( h )) X ϕ So we must have β ( ϕ, f ( g )) = ϕ ( g ), for all ϕ ∈ G . Similarly, β ( ϕ, f ( h )) = ϕ ( h ),for all ϕ ∈ G . Taking ϕ = µ, ν and f ( g ) = µ k ν ℓ , f ( h ) = µ r ν s , we easily find that f ( g ) = ν ℓ where τ ℓ = ω and f ( h ) = µ − r , where τ r = ω − . As a result, we maychoose u ( g ) = X ℓν and u ( h ) = X − ℓµ , where τ ℓ = ω . Proposition 5.10.
Let H be a pointed Hopf algebra of the form T p ( ω ) ⊗ FZ p . If H acts on a matrix algebra A so that the induced grading by G ( H ) is a divisiongrading, then A ∼ = M p ( F ) and the action is isomorphic to an inner action via aconvolution invertible map u : H → A such that for some ≤ ℓ < p and α ∈ F , thematrices for the inner action are given by u ( g ) = X ℓν , u ( h ) = X − ℓµ , where τ ℓ = ω, and u ( x ) = αX µ . For different pairs ( ℓ, α ) the actions are not isomorphic. Actions of H = P ( G , R , D ) and mixed gradings. Let us use the notationand some facts from Section 3. Suppose a Hopf algebra H = P ( G , R , D ) acts on amatrix algebra A . In this case, A becomes a G -graded algebra. In this section wewill look at the action of one of the x i , which we denote by x . We also set a i = a and χ i = χ . Indices i, j will not be used in the sense of Definition 5.1.We set d = d + · · · + d s and view our algebra A as a Kronecker product A = M d ( F ) ⊗ D . The action of g is conjugation by u ( g ), given in (4. To determine u ( x ),we view this matrix as block-diagonal, according to the above splitting of d , withcoefficients in D . Then we can write(37) u ( x ) = X τ ∈T u τ ⊗ X τ and u τ = X ≤ i,j ≤ s u τij , u τij being an ( i, j )th block of u τij . As we know from Theorem 5.4, for any g ∈ G , we must have u ( g ) u ( x ) u ( g ) − = χ ( g ) u ( x ), where χ is the selected character of G . We have u ( g ) u ( x ) u ( g ) − = X σ i ( g ) u τij σ − j ( g ) ⊗ τ ( g ) X τ = χ ( g ) X u τij ⊗ X τ . Since g is arbitrary, the only terms u τij of u ( x ) that survive in (37) are those, forwhich σ i σ − j τ = χ , or σ i = σ j χτ − .We can formulate the result so far obtained, as follows. Proposition 5.11.
Let H , as above, act on a matrix algebra A so that the associ-ated G -grading of A is mixed. We keep the notation used in this section. Then theinner action of H on A can be so chosen that u ( g ) = ( σ ( g ) I d ⊗ X f ( g ) ) ⊕ · · · ⊕ ( σ s ( g ) I d s ⊗ X f ( g ) ) , ∀ g ∈ G ,u ( x ) = X ≤ i,j ≤ s u ij ⊗ X τ ij , if τ ij = σ − i σ j χ ∈ T . In each block row (or column) of u ( x ) there is at most one block different from zero.Proof. Only the last claim needs proof. But if we have σ − i σ j χ ∈ T and σ − i σ k χ ∈ T , then σ − j σ k ∈ T , which is impossible. (cid:3) In some cases more can be said.
Case 1.
An interesting particular case is where χ ∈ T . In this case, only theelements u τii can be nonzero. Also, τ = χ . As a result, we must have u ( x ) = u ⊗ X χ ⊕ · · · ⊕ u ss ⊗ X χ , or u ( x ) = u ′ ( x ) ⊗ X χ , where the grading of u ′ ( x ) ⊗ I equals ε . Subcase (a). Let us first assume that the datum µ from Definition 5.1 equals 0,that is, x n = 0. Then, as we know, we must have u ( x ) n = αI , for some α . If α = 0,then we must also have χ n = ε , and then u nii = αI d i , for all i = 1 , . . . , s . Thus, wemay assume that in this case u ( x ) = u ′ ( x ) ⊗ X χ , where the n th power of u ′ ( x ) isthe scalar matrix. If χ n = ε , this case is not possible. If α = 0 then u ′ ( x ) n = 0. Subcase (b). Let us consider the case where µ = 0. Then we have x n = 1 − a n .We know that in this case, we must have a n = e and χ n = ε . Actually, by Theorem5.5, I + u ( x ) n = λu ( a ) n , for some λ ∈ F . Since χ n = ε , we have u n ⊗ X ε ⊕ · · · ⊕ u nss ⊗ X ε + I d ⊗ X ε = λ (cid:0) σ ( a n ) I d ⊗ X f ( a n ) ⊕ · · · ⊕ σ s ( a n ) I d s ⊗ X f ( a n ) (cid:1) . If f ( a n ) = ε , then, for any τ ∈ T , we must have β ( f ( a n , τ ) = 1 or that X f ( a n ) commutes with any X τ , τ ∈ T . Thus X f ( a n ) = X ε . In this case, we must have u nii = − λσ i ( a ) n I d i , for all 1 ≤ i ≤ s. Thus the n th power of each u ii is a scalar matrix, but this time, if λσ i ( a ) n = 1, thematrix u ii is nilpotent, otherwise it is semisimple.If f ( a n ) = ε then we must have λ = 0 and diag( u , . . . , u ss ) n = − I . This caseis quite similar to Subcase (a), where µ = 0, except that u ′ ( x ) is a nonsingularsemisimple matrix with eigenvalues being n th roots of − Case 2 . Generally, assume that k > χ in G / T . Remember V is a right graded vector space over a graded division algebra D with the support T .We know that u ( x ) can be so chosen that u ( x ) maps V ϕ to V ϕχ , for all ϕ ∈ G .We know that V = L ϕ ∈ G V ϕ . Let W ϕ = L ψ ∈ ϕ T V ψ . Then V = L ϕ ∈ G / T W ϕ .Let us denote by k the order of χ in G / T . Assume k >
1. Choose ϕ ∈ Supp Γ.The sum U ϕ = W ϕ L · · · L W ϕχ k − is direct. We have u ( x ) : W ϕχ i → W ϕχ i +1 , for i = 0 , . . . , k − u ( x ) : W ϕχ k − → W ϕ .By an appropriate choice of bases over D , we can ensure that the matrix for P = u ( x ) on U ϕ is k × k block-diagonal, with only nonzero blocks P i +1 ,i , which arematrices with coefficients in D . Moreover, the blocks P , , P , , . . . , P k,k − can be ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 19 chosen to have all their entries in the base field F . The last matrix P k, is the ma-trix of the map u ( x ) : W ϕχ k − → W ϕ . If { e ( k − , ..., e ( k − d k − } is a basis in W ϕχ − k − ,consisting of the elements of degree ϕχ − k +1 , and { e , ..., e (0) d } , consisting of the ele-ments of degree ϕ , then because, say, deg( u ( x )( e ( k − ) = ϕχ k , and u ( x )( e ( k − ) is alinear combination of homogeneous elements e , ..., e (0) d of degree ϕ with coefficientsin D , all these coefficients must be homogeneous of degree χ k ∈ T , hence equal tothe multiples of X χ k . Thus, we have that all P i +1 ,i are of the forms Y i +1 ⊗ I and P k must have the form of Y ⊗ X χ k .Next we have to take ψ
6∈ { ϕ, ϕχ − , . . . , ϕχ − k +1 } and repeat the process. Finally,the matrix of u ( x ) will be the sum of the blocks, corresponding to U ϕ , where ϕ runsthrough all representatives of the cosets of the subgroup S of G generated by χ and T .Now if we raise the block corresponding to U ϕ , to the power of k , we will obtaindiag { Z ⊗ X χ k , . . . , Z k ⊗ X χ k } , where Z i = Y i Y i +1 · · · Y i − , for all i = 1 , , . . . , k .As earlier, we first assume that the datum µ from Definition 5.1 equals 0, hence, x n = 0. Then, as we know, we must have u ( x ) n = αI , for some α . If α = 0, then wemust also have χ n = ε , and then Z n/ki = αI d i , for all i = 1 , . . . , s . Thus all Z i arenon-singular. Also, in this case, as in Section 5.2, we may assume Y = · · · = Y k = I while Y = Z ⊗ X χ k . As a result, in this case, we have u ( x ) k = diag { Z ⊗ X χ k , . . . , Z ⊗ X χ k | {z } k } . Raising to the power n/k , we find that Z n/k = αI . As a result, in this case ( µ = 0and u ( x ) n = αI , α = 0), if t = | G / S | , the u ( x ) can be reduced to the sum of blocksof the form(38) . . . Z ⊗ X χ k I d t ⊗ X ε . . . . . . . . . . . . . . . . . . . . . I d t ⊗ X ε . Here Z is a matrix, individual for each coset of G / S , such that Z n/k = αI .The determination of the canonical form for u ( x ) in the cases µ = 0 and u ( x ) n =0, as well as in the case µ = 1 is even more technical, so we leave this for a futureresearch. 6. Action of T n ( ω ) on M m ( F )The goal of this section is the classification of the actions of T n ( ω ) on M ( F ).We will apply the results of Section 5.2. Possible forms for u ( x ) are (34) and (33).But we give a more direct treatment here.So we have H = T n ( ω ) and A = M m ( F ) = End V , where ω is an n th primitiveroot of 1 and dim V = n . Since G is cyclic of order n ,generated by an element g , the grading by G on A is elementary, that is, induced from the grading of V = P ϕ ∈ G V ϕ . Now we know that u ( x ) can be chosen so that u ( x ) ∈ A χ where χ ( g ) = ω . Also u ( x ) n = αI , for some α ∈ F . This says that either u ( x ) is semisimplenonsingular or u ( x ) is nilpotent. Now, u ( x )( V ϕ ) ⊂ V ϕχ . So if there is ϕ ∈ Supp V such that ϕχ Supp V then u ( x )( V ϕ ) = { } . In this case, u ( x ) is a nilpotent.Otherwise, u ( x ) is semisimple. As a result, we have two cases, as follows. Case I: V = ⊕ n − i =0 V χ i orCase II: There are ϕ , . . . , ϕ t ∈ G such that V splits as the direct sum of both g - and x -invariant nonzero subspaces V ( i ) = V ϕ i ⊕ V ϕ i χ ⊕ · · · ⊕ V ϕ i χ k − , such that V ϕ i χ k = { } .In Case I, we the matrices for u ( g ) and u ( x ) have the following form: u ( g ) = ϕ ( g ) I d ...
00 ( ϕχ )( g ) I d ... ... ... ... ... ... ( ϕχ n − )( g ) I d n − where ϕ is any element of G .(39) u ( x ) = ... u ,n − u ... u ... ... ... ... .. ... ... ... u n − ,n − We know that u ( x ) n = αI , for some α ∈ F . If α = 0, then arguing as in Section5.2, we will obtain u ( x ) = ... αI d I d ... I d ... ... ... .. ... ... ... I d where d is the common dimension of all V ϕ .We can state it, as follows. Theorem 6.1.
Let u : T n ( ω ) → M m ( F ) define an inner action T n ( ω ) on M m ( F ) suchthat u ( x ) is nonsingular. Then d = nm is an integer and there exists α ∈ F × suchthat the action is isomorphic to the one where u ( g ) = I d ... ωI d ... ... ... ... ... ... ω n − I d , u ( x ) = ... αI d I d ... I d ... ... ... .. ... ... ... I d . Two actions with different values of α are not isomorphic. ✷ Otherwise, if α = 0, we can conjugate u ( x ) in the form (39) by the block di-agonal matrix T = diag { T , . . . , T n } then each block u j +1 ,j will be replaced by T j +1 u j +1 ,j T − j . This allows one to assume we can assume that in the matrix of u ( x ) ( i ) all blocks u j +1 ,j have the form where only the first r j +1 rows are differentfrom zero, where r j +1 is the rank of u j +1 ,j .In Case II, the matrices of u ( g ) and u ( x ) split into the blocks determined by V = ⊕ ti =1 V ( i ) . We will write u ( g ) ( i ) and u ( x ) ( i ) for the i th blocks of u ( g ) and u ( x ). ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 21
Then u ( g ) ( i ) = ϕ i ( g ) I d ...
00 ( ϕ i χ )( g ) I d ... ... ... ... ... ... ( ϕ i χ k − )( g ) I d k − Also, u ( x ) ( i ) = ... u ... u ... ... ... ... .. ... ... ... u k − ,k − Here d , d , . . . , d k − are the dimensions of V ϕ i , V ϕ i χ , . . . , V ϕ i χ k − . If we conjugateby the block diagonal matrix T = diag { T , . . . , T k − } then u ( g ) ( i ) does not changewhile in u ( x ) ( i ) each block u j,j − will be replaced by T j u j,j − T − j − .6.0.1. Actions of T n ( ω ) on M ( F ) . Let A = M ( F ), H = T n ( ω ), n ≥ ω aprimitive n th root of 1. The action of g via matrix u ( g ) = Q makes A a Z n -gradedalgebra, A i = { a | QaQ − = ω i a } . We also set P = u ( x ) and choose P is such away that QP = ωP Q . As a result, P is of degree 1.It easily follows from the isomorphism criterion Proposition 4.2 that we haveonly three nontrivial options for Q : Case 1 Q ( k ) = diag(1 , ω k I ), where 0 < k < n , Case 2 Q ( p, q ) = diag(1 , ω p , ω q ), where 0 < p < q < n .In Case 1 , the trivial component of the grading is the sum of two matrix sub-algebras e Ae ⊕ e Ae , where e = E and e = E + E . Other componentsof the grading are A − k = e Ae and A k = e Ae . So if k = ± n then P = 0. If k = 1 then P = e P e , if k = n −
1, then P = e P e . In both cases,the action will not change, if we conjugate both Q and P by a matrix T of theform T = diag( a, D ), where a ∈ F and D ∈ M ( F ). Computing P ′ = T P T − , weobtain P ′ = e ( aP D − ) e , in the first case, and P ′ = e ( aDP ) e in the second.Clearly, we can choose T so that P gets the form P (1) = E , and in the second, P (2) = E . Case 2 . This time, the components of the grading are A = Span { E , E , E } ,.Other components are A p = Span { E } , A − p = Span { E } , A q = Span { E } , A − q = Span { E } , A p − q = Span { E } , A q − p = Span { E } . If none of the num-bers ± p , ± q , ± ( p − q ) is congruent 1 mod n , we have to set P = 0.We can write P as follows: P = αE + βE + γE . We can conjugate P by any nonsingular diagonal matrix T = diag( a, b, c ). We have P ′ = T P T − =( a/b ) αE + ( b/c ) βE + ( c/a ) γE . If one or two of the numbers α, β, γ equalzero then the remaining numbers can be made equal 1, which gives 6 pairwisenonisomorphic options for P : P (3) = E , P (3) = E , P (3) = E , P (3) = E + E ,P (3) = E + E , P (3) = E + E . In the case where all three of the numbers α, β, γ are nonzero, which is only possibleif n ≤
3, the best we can do is to reduce P to P (3) γ = E + E + γE where γ isa nonzero number. Since P (3) = γI , for different values of γ the actions are notequivalent. Theorem 6.2. If n = 3 then any action of T n ( ω ) on M ( F ) is isomorphic to oneof the inner actions via u : H → M ( F ) where the pair ( u ( g ) , u ( x )) is one of thefollowing ( Q (1) , P (1)) , ( Q ( n − , P (2)) , ( Q (1 , n − , P (3) i ) , i = 1 , . . . . If n = 3 ,then we can also have ( Q (1 , , P (3) γ ) , γ ∈ F × . All these actions are pairwisenon-isomorphic. Actions of D ( T n ( ω )) on M m ( F )The Drinfeld double of a Taft algebra is a particular case of algebras P ( G , R , D )defined in Section 5. We will use the following presentation of the Drinfeld double H = D ( T n ) of the Taft algebra T n ( ω ), in the case where n is an odd number > H is generated by g, x, G, X , where g, G are group-likes and x, X are skew-primitives and g n = G n = 1 , x n = X n = 0 ,gG = Gg, gx = ω − xg, Gx = ω − xG, gX = ωXg, GX = ωXG,xX − ωXx = 1 − gG ∆( x ) = x ⊗ g ⊗ x, ∆( X ) = X ⊗ G ⊗ X. Generally, it is possible in the definition of P ( G , R , D ) to replace some of thevariables x i to x i a − i and a i to a − . In that case, the relations (18) between an“old” and “new” x i , x j will change to x j x i − x i x j = g − G . The relations betweentwo “old” and two ‘new” variables will looks the same way. In particular, this canbe done with the generators of the Drinfeld double to get xX − Xx = G − g . Inthe next result we will show the connection between elements of the inner actionwhen this change of “old” to “new” takes place. Proposition 7.1.
Let u : H → A provides an inner action of a Hopf algebra H ona central algebra A . Suppose that ∆( x ) = x ⊗ a + 1 ⊗ x is a ( a, -primitive elementof H , a a grouplike. Let x = xa − . Then ∆( x ) = x ⊗ a − + a − ⊗ x and (40) u ( x ) = u ( x ) u ( a ) + λI and u ( x ) = u ( x ) u ( a ) − − λu ( a ) , for some λ ∈ F . Proof.
Let A be an arbitrary element of A ; we compute the action of x on A intwo different ways, using formulas (8,9,10). Then x ∗ A = u ( x ) Au ( a ) − − Au ( x ) u ( a ) − , x ∗ A = u ( x ) A − u ( a ) − Au ( a ) u ( x ) . Since x = x a , we have x ∗ A = x ∗ ( u ( a ) Au ( a ) − ) = u ( x ) u ( a ) Au ( a ) − − u ( a ) − u ( a ) Au ( a ) − u ( a ) u ( x )As a result, we have u ( x ) Au ( a ) − − Au ( x ) u ( a ) − = u ( x ) u ( a ) Au ( a ) − − Au ( a ) − u ( a ) u ( x ) . Then ( u ( x ) − u ( x ) u ( a )) Au ( a ) − = A ( u ( x ) u ( a ) − − u ( a ) − u ( a ) u ( x )) or( u ( x ) − u ( x ) u ( a )) A = A ( u ( x ) − u ( x ) u ( a )) . Since A is central, there is λ ∈ F , such that u ( x ) − u ( x ) u ( a ) = λI. From this equation both equations (40) are immediate. (cid:3)
ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 23
The above proposition is important because in distinction from Definition 5.1,where all skew primitive elements are (1 , a )-primitive, in some “real life” example,like algebras of dimension p in [1], some skew-primitive elements are (1 , a )-primitiveand some are ( b, λu ( a ) − and λI in (40) can be ignored when computing the action of x and y . This factwas proved in Lemma 4.1.7.1. Inner actions of D ( T n ) . If A is an H -algebra then we have g ∗ G ∗ x ∗ X ∗ a, b ∈ A , we have g ∗ ( ab ) = ( g ∗ a )( g ∗ b ) , G ∗ ( ab ) = ( G ∗ a )( G ∗ b ) ,x ∗ ( ab ) = ( x ∗ a ) b − ( g ∗ a )( x ∗ b ) , X ∗ ( ab ) = ( X ∗ a ) b − ( G ∗ a )( X ∗ b ) . If A is a matrix algebra, there is a map u : H → A such that, for any a ∈ A , wehave h ∗ a = u ( h ) au ( h ) − , for any h ∈ G = G ( H ) ,x ∗ a = u ( x ) a − u ( g ) au ( g ) − u ( x ) , X ∗ a = u ( X ) a − u ( G ) au ( G ) − u ( X ) . In the case, where the group G = G ( H ) of grouplikes of a Hopf algebra H actingon a matrix algebra A = M m ( F ) is not cyclic we need to consider the gradings onthe matrix algebras which do not need to be elementary.Now G = G ( H ) = ( g ) n × ( G ) n ∼ = Z n × Z n . The dual group G consists of pairs( χ, ψ ) of characters χ ∈ c ( g ), ψ ∈ d ( G ). Each such pair can be identified with thepair ( k, ℓ ) of numbers modulo n . We have a ∈ A k,ℓ if and only if g ∗ a = ω k a , G ∗ a = ω ℓ a . It follows from the defining relations that x ∗ A k,ℓ ⊂ A k − ,l +1 and X ∗ A k,ℓ ⊂ A k +1 ,l − . For example, if the action of ( g ) is not faithful then theaction of both x and X is trivial. The proof of Lemma 3.2 works both for x and X . Similarly, with G in place of g . At the same time, it is possible that the actionof gG is trivial and x , X still act in a non-trivial way. Actually, in this case onespeaks about the actions of H = u q ( sl ) (see Section 8). Note that in this case,according to Proposition 3.1, the support of the grading is a subgroup of G equal tothe annihilator of gG , which is a cyclic subgroup isomorphic to Z n . So the gradingslifted from H = u q ( sl ) must be elementary.The following proposition is a direct consequence of Theorems 5.3, 5.5, 5.6, Proposition 7.2. If H = D ( T n ) acts on a matrix algebra A then the action isinner and the matrices of the inner action can be chosen so that (41) u ( g ) u ( x ) u ( g ) − = ω − u ( x ) , u ( G ) u ( x ) u ( G ) − = ω − u ( x ) , (42) u ( g ) u ( X ) u ( g ) − = ωu ( X ) , u ( G ) u ( X ) u ( G ) − = ωu ( X ) . and (43) u ( x ) u ( X ) − ωu ( X ) u ( x ) = I + λu ( g ) u ( G ) for some constant λ ∈ F . If the accompanying grading by G is not elementary then (44) u ( x ) u ( X ) − ωu ( X ) u ( x ) = I. Proof.
It follows from (44) that if the grading is not elementary then all matriceson the left hand side, as well as I have degree ε , while u ( g ) u ( G ) has degree differentfrom ε (see (4). As a result, in this case, we must have a more restrictive relation(44). (cid:3) Let us define a character χ : G → F × by setting χ ( g ) = ω − , χ ( G ) = ω .The sufficient conditions for the action of D ( T n ) on M m ( F ) will now look like u ( g ) n = I m , u ( G ) n = I m ,u ( h ) u ( x ) = χ ( h ) u ( x ) u ( h ) , u ( h ) u ( X ) = χ − ( h ) u ( X ) u ( h ) , for any h ∈ G u ( x ) n = ζI, u ( x ) n = ξI, ζ, ξ ∈ F u ( x ) u ( X ) − ωu ( X ) u ( x ) = I + λu ( g ) u ( G ) , λ ∈ F . An explicit form for of the division actions of D ( T n ( ω )) can be given, as follows.First, if x and X act nontrivially, then by Theorem 3.1, the support of the gradingcannot be a proper subgroup of G . So if G = ( µ ) × ( ν ), o ( µ ) = o ( ν ) = n , then also T = ( µ ) × ( ν ). We also need to fix an alternating bicharacter β : T × T → F × . Thisis done by fixing a primitive n th root of 1, say π . Now, as we concluded above, u ( x ) = γX − χ while u ( X ) = βX χ , for some γ, δ ∈ F . It follows from (44) that u ( x ) u ( X ) − ωu ( X ) u ( x ) = γδ (1 − ω ) .I = I. Thus we must have γδ = 11 − ω .Now to determine u ( g ) and u ( G ), we should act exactly as in Section 5.3.1. Easycalculations show that we must have u ( g ) and u ( G ) be scalar multiples of u ( g ) = X − ℓν and u ( G ) = X ℓµ where ℓ is uniqely defined by π ℓ = ω. One easily checks, for example, u ( G ) u ( x ) u ( G ) − = ( X ℓ − µ )( γX − χ )( X ℓµ ) = β ( µ − ℓ , χ − ) X − χ = β ( µ ℓ , µ − ν − ) X − χ = β ( µ, ν ) − ℓ X − χ = π − ℓ X − χ = ω − u ( g ) . An explicit example for n = 3 looks like the following. Fix a 3rd primitive root ω of 1 and consider two options. If π = ω then u ( g ) = , u ( G ) = ω
00 0 ω ,u ( x ) = γ ω ω , u ( X ) = δ ω ω . If π = ω then ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 25 u ( g ) = , u ( G ) = ω
00 0 ω ,u ( x ) = γ ωω , u ( X ) = δ ω ω . We also must have γδ = 11 − ω .7.2. Mixed actions of D ( T ) . In this section we will use the following notationfor the Sylvester/Pauli matrices in the case n = 2: A = (cid:18) − (cid:19) , B = (cid:18) (cid:19) , C = (cid:18) − (cid:19) . Then, if D ( T ) acts on R = M k ( F ) such that the grading by G is mixed, thematrices of the inner action can be chosen as follows: u ( g ) = I k ⊗ A, u ( G ) = I k ⊗ B. The commutation relations (41) and (42) will now take the form u ( g ) u ( x ) = − u ( x ) u ( g ) , u ( G ) u ( x ) = − u ( x ) u ( G ) ,u ( g ) u ( X ) = − u ( X ) u ( g ) , u ( G ) u ( X ) = − u ( X ) u ( G ) . It is now easy to compute the matrices u ( x ) and u ( X ):(45) u ( x ) = P ⊗ C, u ( X ) = Q ⊗ C for some P, Q ∈ M k ( F ) . Using (44), we get the following I k = u ( x ) u ( X )+ u ( X ) u ( x ) = ( P ⊗ C )( Q ⊗ C )+ Q ⊗ C )( P ⊗ C ) = − ( P Q + QP ) ⊗ I . (46) P Q + QP = − I k . Note that if T ∈ GL k then one can conjugate R by T ⊗ I without changing thegrading. This allows one to reduce, say P to a canonical form P = T P T − andfind Q = T QT − from the equation P Q + Q P = − I k .One also needs to remember that ( P ⊗ C ) = − P ⊗ I and ( Q ⊗ C ) = − Q ⊗ I are scalar matrices. So in our future calculations we may assume that P and Q are matrices satisfying (46), their squares are scalar matrices and P is in thenormal Jordan form. Case 1 : One of
P, Q is nilpotent, say P = 0. Set J = (cid:18) (cid:19) . Then we cantake P in the form P = (cid:18) P ′
00 0 (cid:19) where P ′ = diag { J, . . . , J | {z } r } . Easy calculation using (46) show that P ′ must be the whole of P . It will be moreconvenient to write(47) P = (cid:18) I r (cid:19) where k = 2 r. In the above formula zeroes are zero r × r -matrices. We also write Q = (cid:18) Q Q Q Q (cid:19) where each Q ij is an r × r -matrix. If we plug P and Q , as just above, to (46), wewill obtain (cid:18) Q Q + Q Q (cid:19) = (cid:18) − I r − I r (cid:19) . Then Q = (cid:18) Q Q − I r − Q (cid:19) . To further simplify u ( X ) = Q ⊗ C , we can conjugate this matrix by any matrixwhose conjugation maps u ( g ) = I k ⊗ A and u ( G ) = I k ⊗ B to their scalar multi-ples (by ±
1) and u ( x ) to u ( x )+ µu ( g ) (see Lemma 4.2). If one writes the conjugatingmatrix as T = T I ⊗ I + T A ⊗ A + T B ⊗ B + T C ⊗ C, then the condition T ( I k ⊗ A ) T − = ± I k ⊗ A and T ( I k ⊗ B ) T − = ± I k ⊗ B implies that T must be one of T I ⊗ I, T A ⊗ A, T B , ⊗ B, T C ⊗ C . If we now conjugate u ( x ) = P ⊗ C by one of these matrices, then, first of all we must have µ = 0 andalso, depending on what matrix has been chosen, T I P = P T I , T A P = − P T A , T B P = − P T B and T C P = P T C .In the case where T P = P T , using (47), we find that T = (cid:18) T T T (cid:19) . In thecase T P = − P T , we have T = (cid:18) T T − T (cid:19) . Here T ∈ GL r and T ∈ M r ( F ).Thus in both cases we can conjugate first by a matrix where T = I r and theconjugate by T = (cid:18) T ± T (cid:19) .Before we do this, we remember that Q = αI k for some α ∈ F . Hence, αI k = (cid:18) Q − Q Q Q − Q Q Q − Q (cid:19) . As a result, Q = Q − αI r and Q = (cid:18) Q Q − αI r − I r − Q (cid:19) . If we conjugate this matrix by T = (cid:18) I r T σI r (cid:19) , where σ = ±
1, we will obtain thefollowing
T QT − = (cid:18) Q − T − σQ T + σT + σQ − ασI r − σT Q − σI r T − Q (cid:19) . If we set T = Q then T QT − = (cid:18) − ασI r − σI r (cid:19) . Proposition 7.3. If D ( T ) acts on M m ( F ) , so that u ( g ) does not commute with u ( G ) and u ( x ) is nilpotent then m = 2 k and the action is isomorphic to preciselyone inner action where u ( g ) = I k ⊗ A, u ( G ) = I k ⊗ B, u ( x ) = (cid:18) I r (cid:19) ⊗ C, u ( X ) = (cid:18) τ I r − I r (cid:19) ⊗ C, ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 27 for some τ ∈ F .Proof. This follows if in the preceding argument we use conjugation by a matrix T ⊗ I where T = (cid:18) I r T I r (cid:19) . Also, our argument about possible shapes for theconjugating matrix T shows that different τ produce non-isomorphic actions. (cid:3) Case 2 : P = αI k , Q = βI k , α, β = 0. Choose ξ ∈ F such that xi = α . Thenafter conjugation by a matrix T ⊗ I , which does not change the actions by u ( g )and u ( G ), we can reduce P to P = (cid:18) ξI r − ξI s (cid:19) . Let apply (46) to P as justabove and Q = (cid:18) X YZ U (cid:19) , where the blocks have the sizes r × r , r × s , s × r , and s × s . Then (cid:18) ξI r − ξI s (cid:19) (cid:18) X YZ U (cid:19) + (cid:18) X YZ U (cid:19) (cid:18) ξI r − ξI s (cid:19) = (cid:18) − I r − I s (cid:19) . In this case, 2 ξX = − I r , 2 ξY = − I s , and we can assume Q = − ξ (cid:18) I r YZ − I s (cid:19) . Let us also remember Q = βI k , were β = 0. Then βI m = 14 α (cid:18) I r + Y Z ZY + I s (cid:19) , hence Y Z = (4 αβ − I r , ZY = (4 αβ − I s . Unless 4 αβ − r = s and Y Z = ZY = 14 αβ − I r . In thislatter case, any further reductions can only be provided by the matrices of the form T = (cid:18) K L (cid:19) or U = T (cid:18) I r I r (cid:19) . We have(48) (cid:18) K L (cid:19) (cid:18) I r YZ − I r (cid:19) (cid:18) K − L − (cid:19) = (cid:18) I r KY L − LZK − − I r (cid:19) . One can choose K and L so that KY L − = I r . Now, in general, ( LZK − )( KY L − ) =14 αβ − I r . Thus, with the latter choice of K and L , we should have and so( LZK − ) = 14 αβ − I r and Q = (cid:18) I r I r αβ − I r − I r (cid:19) . As a result, in the case where4 αβ = 1, we have k = 2 r and, for a choice of xi with xi = α , u ( g ) = I k ⊗ A, u ( G ) = I k ⊗ B, u ( x ) = ξ (cid:18) I r − I r (cid:19) ⊗ C,u ( X ) = (cid:18) I r I r αβ − I r − I r (cid:19) ⊗ C. It is easy to see that for different choices of of ξ and β , the actons parametrizedby these constants are not isomorphic (remember α = ξ ).Now let us consider the remaining case where Y Z = ZY = 0. Using our cal-culation in (48), which is true also when r = s , we will reduce Q to the form Q = (cid:18) I r KY L − LZK − − I s (cid:19) where now(49) ( KY L − )( LZK − ) = ( KY L − )( LZK − ) = 0 . For appropriate K and L , we can assume KY L − = (cid:18) I t t,s − t r − t,t r − t,s − t (cid:19) , for some t ≤ r, s . From (49), we then find LZK − = (cid:18) t,t t,r − t s − t,t Ξ (cid:19) , where Ξ is a rectangu-lar ( s − t ) × ( r − t )-matrix. In this case, we still have ξ as one of the parameters of theaction, but instead of one parameter β , we have an ( s − t ) × ( r − t )-array of parametersΞ. So if we set Y ( r, s, t ) = (cid:18) I t t,s − t r − t,t r − t,s − t (cid:19) and Z ( r, s, Ξ) = (cid:18) t,t t,r − t s − t,t Ξ (cid:19) ,then the last family of the actions is given by u ( g ) = I k ⊗ A, u ( G ) = I k ⊗ B, u ( x ) = ξ (cid:18) I r − I s (cid:19) ⊗ C,u ( X ) = (cid:18) I r Z ( r, s, Ξ) Y ( r, s, t ) − I s (cid:19) ⊗ C. Proposition 7.4. If D ( T ) acts on M m ( F ) , so that u ( g ) does not commute with u ( G ) and u ( x ) is not nilpotent then m = k and the action is isomorphic to an inneraction where u ( g ) = I k ⊗ A, u ( G ) = I k ⊗ B, u ( x ) = ξ (cid:18) I r − I s (cid:19) ⊗ C,u ( X ) = (cid:18) I r Z ( r, s, Ξ) Y ( r, s, t ) − I s (cid:19) ⊗ C. Two such actions, with parameters r, s, t, ξ, Ξ and r ′ , s ′ , t ′ , ξ ′ , Ξ ′ , are isomorphicif and only if r = r ′ , s = s ′ , t = t ′ , ξ = ξ ′ and there is U ∈ GL s ( F ) such that Ξ ′ = U Ξ U − . Thus, we have obtained a full classification of non-elementary actions of D ( T )on M n ( F ), where F is an algebraically closed field of characteristic zero.7.3. General elementary actions of the Drinfeld double.
We now examinethe elementary actions of the Drinfeld doubles D ( T n ( ω )) in the case n ≥ A = End V , V a finite-dimensional vector space over the basefield F , containing n th roots of 1. Let m = dim V . Note that by (43) we always have u ( x ) u ( X ) − ωu ( X ) u ( x ) = I + λu ( g ) u ( G ). Because the grading by G is elementary,we have(50) V = ⊕ ϕ ∈ G V ϕ and u ( h ) = X ϕ ∈ G ϕ ( h ) π ϕ for any h ∈ G . Here we denote by π ϕ the projection of V onto V ϕ .Now let χ ∈ G be a uniquely defined character such that χ ( g ) = χ ( G ) = ω − Then, as we know, u ( h ) u ( x ) u ( h ) − = χ ( h ) u ( x ) and u ( h ) u ( X ) u ( h ) − = χ − ( h ) u ( X ) , ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 29 for all h ∈ G . It then follows that u ( x ) ∈ A χ and u ( X ) ∈ A χ − . Any gradedcomponent A δ of A consists of linear transformations a , satisfying a ( V ψ ) ⊂ V δψ ,for any ψ ∈ G . If T is the cyclic subgroup of order n generated by χ , the space V is split as the sum of subspaces V ¯ ϕ , ¯ ϕ ∈ G / T , which are invariant under the actionof u ( g ) , u ( G ) , u ( x ) , u ( X ). We have V ¯ ϕ = X ψ ∈ ¯ ϕ V ψ If we fix any ϕ ∈ ¯ ϕ and set V kϕ = V ϕχ k , for k = 0 , , . . . , n −
1, then u ( x ) : V kϕ → V k +1 ϕ and u ( X ) : V kϕ → V k − ϕ . We know that there exist α, β ∈ F such that u ( x ) n = αI m , u ( X ) n = βI m .In what follows we restrict ourselves to the case when one of u ( x ), u ( X ) isnonsingular. Let us assume α = 0. In this case, as we’ve seen earlier in Section 5.2,for any ϕ ∈ b G fixed, the dimensions of all V kϕ are the same. Let us set f k = π ϕ k +1 ◦ u ( x ) ◦ π ϕ k and g k = π ϕ k − ◦ u ( X ) ◦ π ϕ k . Now f k − ◦ · · · ◦ f k +1 ◦ f k = α π ϕ k . So all f k are isomorphisms. Let dim V kϕ = r . If we choose a basis B in V ϕ and set B k = f k ( B k − ), for any k = 1 , , . . . , n −
1, then we will obtain a basis B = B ∪ . . . ∪ B n − such that the matrix for u ( x ) | V ¯ ϕ will have the form · · · αI r I r · · · I r · · · · · · · · · · · · · · · · · · · · · I r . To determine u ( X ), note that the subspaces V ¯ ϕ are invariant under all u ( x ), u ( X ), u ( g ), u ( G ). This allows us to use the relations for these matrices derived earlier,for the restrictions to each of these subspace. For instance, we will have g k +1 ◦ · · · ◦ g k − ◦ g k = β π ϕ k . Let us use u ( x ) u ( X ) − ωu ( X ) u ( x ) = I − λu ( g ) u ( G ), restricted to V ¯ ϕ . We have n − X k =0 ( f k − ◦ g k − ωg k +1 ◦ f k ) = n − X k =0 (1 − λϕχ k ( gG )) π ϕ k or n − X k =0 ( f k − ◦ g k − ωg k +1 ◦ f k ) = n − X k =0 (1 − λω − k ϕ ( gG )) π ϕ k Let us set ρ k = (1 − λω − k ϕ ( gG )). Then for each k = 0 , , . . . , n −
1, we willhave f k − ◦ g k − ωg k +1 ◦ f k = ρ k π ϕ k . This relation allows us to express all g k in terms of one of them, once f , . . . , f k − is known. For any k = 0 , , . . . , n − g k = ωf − k − ◦ g k +1 ◦ f k + ρ k f − k − . Using the above basis B for V ϕ , let us denote by ( v ij ) the block diagonal matrixfor u ( X ) in this basis and ( u ij ) the matrix for u ( x ). We have u ( X ) = v · · ·
00 0 v · · · · · · · · · · · · · · · · · · · · · v n − ,n v n · · · . Then v k − ,k = ωu − k,k − v k,k +1 u k +1 ,k + ρ k u − k,k − . We already know that the only nonzero blocks in this matrix are u k +1 ,k , for k =0 , . . . , n −
1, and u k +1 ,k = I r if k = 1 , . . . , n − u ,n = αI r . As a result, wewill obtain the recurrent matrix relation v k − ,k = ωv k,k +1 + ρ k I r . for k = 1 , . . . , n − v n − , = α − ( ωv , + ρ n I r ) v n − ,n − = α ( ωv n − , + ρ n − I r )This recurrent relation enables us to express all blocks of the matrix of therestriction u ( X ) to V ¯ ϕ by a single one, say, v ,n − . Moreover, if we conjugate allmatrices in question by any diag { T, T, . . . , T } , where T is a non-singular squarematrix of order r , then the matrices for u ( g ) , u ( G ) , u ( x ) remain the same while wemay assume that v ,n − is a matrix in a Jordan normal form.8. Actions of H = u q ( sl )We adopt the following presentation of H = u q ( sl ) from [1]. H is generated bythe elements a, x, y such that a n = 1 , x n = y n = 0 ax = ω xa, ay = ω − ya, xy − yx = a − a − . (51)Here n is an odd number, n ≥ x = x ⊗ a + 1 ⊗ x, ∆ y = y ⊗ a − ⊗ y, ∆ a = a ⊗ a. Here ω is a primitive n th root of 1, n an odd number.Suppose H acts on a matrix algebra A . Then the action is inner, via a convo-lution invertible map u : H → A and given by the formulas below, where A is anyelement of A . a ∗ A = u ( a ) Au ( a ) − x ∗ A = u ( x ) Au ( a ) − − Au ( x ) u ( a ) − y ∗ A = u ( y ) A − u ( a ) − Au ( a ) u ( y )(52) Proposition 8.1.
The elements u ( a ) , u ( x ) , u ( y ) ∈ A on the right hand sides of theequations in (52) can be so chosen that (53) u ( a ) n = θ , (54) u ( a ) u ( x ) = ω u ( x ) u ( a ) , ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 31 (55) u ( a ) u ( y ) = ω − u ( y ) u ( a ) , (56) u ( x ) n = µ , u ( y ) n = ν u ( x ) u ( y ) − u ( y ) u ( x ) = u ( a ) − τ u ( a ) − , where θ = 0 , τ, µ, ν ∈ F .Proof. Note that actually H is an algebra of the type described in the Definition5.1, if we set G = ( a ), x = xa − , a = a − , x = y , a = a − , χ ( a ) = ω , χ ( a ) = ω − . We also have N = N = n , µ = µ = 0, hence x n = x n = 0. Also, x x − χ ( a ) x x = yxa − − χ ( a ) − xa − y (58) = yxa − − ω − xa − y = yxa − − xya − (59) = ( a − − a ) a − = − (1 − a a )(60)It follows that λ = −
1. Notice also that χ ( a ) χ ( a ) = 1, as needed inDefinition 5.1. Now one can apply Theorems 5.3, 5.5, 5.6 to get our needed relations(53), (54), (55), (56), (57). The main tool here is Proposition 7.1.For example, for any g ∈ G , we have u ( g ) u ( x ) u ( g ) − = χ ( g ) u ( x ) so that u ( a ) u ( x ) u ( a ) − = ω u ( x )Using Proposition 7.1, we get u ( a ) u ( x ) u ( a ) − − λu ( a ) − = ω u ( x ) u ( a ) − − ω λu ( a ) − u ( a ) u ( x ) u ( a ) − = ω u ( x ) + λ (1 − ω ) I, as claimed.As another example, u ( x ) u ( x ) − χ ( a ) u ( x ) u ( x ) = − ζ u ( a ) u ( a ) u ( y ) u ( x ) u ( a ) − − ω − u ( x ) u ( a ) − u ( y ) = − ζ u ( a ) − u ( y ) u ( x ) − u ( x ) u ( y ) = − u ( a ) + ζ u ( a ) − u ( x ) u ( y ) − u ( y ) u ( x ) = u ( a ) − ζ u ( a ) − , as needed. (cid:3) Book Hopf algebra.
One more Hopf algebra, which is a part of the classi-fication of Hopf algebras of dimension p , p is an odd prime number, in [1], is theso called Book Hopf algebra h ( q, m ), where q is p -th root of 1, and m ∈ Z p \ { } .In terms of generators and defining relations generators and defining relations thisalgebra is given by h ( q, m ) = h a, x, y | axa − = qx, aya − = q m y, a p = 1 , x p = 0 , y p = 0 , xy − yx = 0 i . The coproduct is given by the following∆( a ) = a ⊗ a, ∆( x ) = x ⊗ a + 1 ⊗ x, ∆( y ) = y ⊗ a m ⊗ y. The following is very similar by the statement and the proof to Proposition 8.1.
Suppose H acts on a matrix algebra A . Then the action is inner, via a convo-lution invertible map u : H → A and given by the formulas below, where A is anyelement of A . a ∗ A = u ( a ) Au ( a ) − x ∗ A = u ( x ) Au ( a ) − − Au ( x ) u ( a ) − y ∗ A = u ( y ) A − u ( a ) m Au ( a ) − m u ( y )(61) Proposition 8.2.
The elements u ( a ) , u ( x ) , u ( y ) ∈ A on the right hand sides of theequations in (61) can be so chosen that (62) u ( a ) n = θ , (63) u ( a ) u ( x ) = qu ( x ) u ( a ) , (64) u ( a ) u ( y ) = q m u ( y ) u ( a ) , (65) u ( x ) n = µ , u ( y ) n = ν u ( x ) u ( y ) − u ( y ) u ( x ) = τ u ( a ) m , where θ = 0 , τ, µ, ν ∈ F .Proof. Again, in the same way as in Proposition 8.1, we note that the algebra inquestion is an algebra from Definition 5.1. After this, our relations (62), (63), (64),(65), (66) follow with the use of Theorems 5.3, 5.5, 5.6 and Proposition 7.1. (cid:3)
Actions on M ( F ) . As an example of the results in the previous section, weconsider the action of H = u q ( sl ), on A = M ( F ). First of all, there is a basisof the underlying vector space F where u ( a ) = λ (cid:18) ω k (cid:19) , for λ ∈ F satisfying λ n = 1, 1 ≤ k < n . If u ( x ) = τ (cid:18) a bc d (cid:19) , then u ( a ) u ( x ) u ( a ) − = ω u ( x ) , that is (cid:18) a ω n − k bω k c d (cid:19) = (cid:18) ω a ω bω c ω d (cid:19) . Since we assume n ≥
3, we must have a = d = 0. We also must have ω n − k b = ω b and ω k c = ω c . Thus if none of k , n − k equals 2, the action of x is trivial (thisalso follows by Proposition 3.1). Since the computation for u ( y ) is similar, in thecase none of k, n − k equals 2, the action of H is purely a group action, so is justa grading by G ∼ = Z n . Since also n = 4, only one of k , n − k equals 2. Let us firstlook at the case k = 2. In this case u ( x ) = (cid:18) p (cid:19) , u ( y ) = (cid:18) q (cid:19) , for some p, q ∈ F . Clearly they satisfy all (53) to (56), so we need only to see whatrestrictions are imposed by (57). We have (cid:18) − qp pq (cid:19) = (cid:18) λ − λ − τ λω − λ − τ ω n − (cid:19) Thus τ = λ ω ω n − and pq = λ ω − ω n − ω n − . Similar is the case n − k = 2. ROUP GRADINGS AND ACTIONS OF POINTED HOPF ALGEBRAS 33 If n = 4 then k = 2 and we have u ( x ) = (cid:18) pq (cid:19) , u ( y ) = (cid:18) rs (cid:19) , for some p, q, r, s ∈ F . Again, we need to check (57). It follows from (57) that (cid:18) ps − qr qr − ps (cid:19) = ( λ − λ − τ ) (cid:18) ω (cid:19) , Since ω = −
1, it follows that any u ( x ) , u ( y ) can be chosen, and (57) will besatisfied with τ = λ − ps + qr . This enough freedom to produce different actions of H on A . It would be good to determine the equivalence classes of actions.8.3. Actions of Drinfeld doubles of the Taft algebra via the actions of u q ( sl ) . An alternate approach to the results in Section 7 can be given as follows(see [15, Corollary 4.8], we use the notation from [12, IX.6]). First, it is well-knownthat u q ( sl ) is a Hopf homomorphic image of D ( H ) , for H = T n ( ω ) , by setting ω = q and defining Φ : D ( H ) → u q as follows: G K , g K − , x F , and X EK − := − ( q − q − ) E .Moreover Ker (Φ) = D ( H ) k G + , where G is the cyclic subgroup generated by gG . Corollary 8.3.
Each action of u q ( sl ) on an algebra A lifts to an action of theDrinfeld double D ( T q ) on A via the homomorphism shown above. Thus any actionof u q ( sl ) on M n ( F ) can be obtained from an elementary action of D ( T q ) describedin Section 7.3, with an additional provision that in (50) the summation is takenover the characters ϕ ∈ G such that ϕ ( gG ) = 1 . Note that the converse is not true: no division action of the Drinfeld double D ( T q ) comes as a lifting of an action of the u q ( sl ), because G and g do not act asinverses of each other. For more specific examples see the formulas at the end ofSection 7.1). References [1] Andruskiewitsch, N.; Schneider, H.-J.,
Lifting of quantum linear spaces and pointed Hopfalgebras of order p , J. Algebra (1998), 658–691.[2] Angiono, I.; Garcia Iglesias, A., Pointed Hopf algebras: a guided tour to the liftings , Rev.Colomb. Math., (2019), 1–44.[3] Bahturin, Y.; Kochetov, M., Classification of group gradings on simple Lie algebras of types A , B , C , and D , J. Algebra (2010), 2971–2989.[4] Bahturin,Y.; Sehgal, S; Zaicev, M., Group gradings on associative algebras , J. Algebra (2001), 667–698.[5] Bahturin, Y.; Zaicev, M.,
Group gradings on matrix algebras , Canad. Math. Bull., (2002),499–508.[6] Bahturin, Y.; Zaicev, M., Graded algebras and graded identities , Polynomial identities andcombinatorial methods (Pantelleria, 2001), Lecture Notes in Pure and Appl. Math., (2003), 101–139.[7] Carnovale, G.; Cuadra, J.,
On the subgroup structure of the full Brauer group of SweedlerHopf algebra , Israel J. Math., (2011), 61–92.[8] Cuadra, J.; Etingof, P.,
Finite dimensional Hopf actions on central division algebras , Int.Math. Res. Not. IMRN 2017, no. 5, 1562–1577[9] Elduque, A.; Kochetov, M., Group gradings on simple Lie algebras, AMS Math. Surv. Mon. (213), xiii+336pp.[10] Etingof, P.; Walton, C.,
Semisimple Hopf actions on commutative domains , Adv. Math. (2014), 4761. [11] Gordienko, A.,
Algebras simple with respect to a Taft algebra action , J. Pure Appl. Algebra (2015), 3279–3291.[12] Kassel, C., Quantum Groups, Graduate Texts in Mathematics, 1 , Springer-Verlag, NewYork, 1995, xii+531 pp.[13] Krop, L.; Radford, D., Finite-dimensional simple-pointed Hopf algebras , J. Algebra (1999) 686–710.[14] Masuoka, A.,
Coalgebra actions on Azumaya algebras , Tsukuba J. Math. (1990), 107–112.[15] Montgomery, S., Hopf algebras and their actions on rings, CBMS Regional Conference Seriesin Mathematics, , American Mathematical Society, Providence, RI, 1993. xiv+238 pp.[16] Montgomery, S.; Schneider, H.-J., Skew derivations of finite-dimensional algebras and ac-tions of the double of the Taft Hopf algebra , Tsukuba J. Math. (2001), 337–358.[17] Nˇastˇasescu, C.; Van Oystaeyen, F., Methods of graded rings, Lecture Notes in Mathematics, (2004), 306pp.[18] Radford, D., Finite-dimensional simple-pointed Hopf algebras , J. Algebra (1999), 686–710.[19] van Oystayen, F.; Zhang, Y.,
The Brauer group of Sweedler’s Hopf algebra H , Proc. AMS (2000), 371–380. Department of Mathematics and Statistics, Memorial University of Newfoundland,St. John’s, NL, A1C5S7, Canada
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