Group gradings and actions of pointed Hopf algebras

Group gradings and actions of pointed Hopf algebras

Yuri Bahturin and Susan Montgomery

LieJor Seminar São Paulo May 12, 2022

Figure:“Graded Algebras and Superalgebras”, Memorial University, 2008

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Figure:“Graded Algebras and Superalgebras”, Memorial University, 2008

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Many papers have been devoted to the study of actions of Hopf algebras on algebras. However very few of them have actually classified the possible actions of a certain Hopf algebra on a given algebra. One of the first of these papers was the determination of the actions of the Taft algebra and its double on a 1-generator 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 been studied, in order to look at their polynomial identities.

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In this paper we study the actions of finite-dimensional pointed Hopf algebras with abelian group of group-like elements on matrix algebras. Our methods are novel, in that we use very strongly the classification of group gradings on matrices, as in [4].

LetH be a (finite-dimensional) Hopf algebra over a field F. Suppose thatH acts on a unital algebraA. This means that there is a unital leftH-module structure on Awritten by (h,a)→h∗a such thath∗1 =ε(h)1 andh∗(ab) = (h₍₁₎∗a)(h₍₂₎∗b) where h∈H,a,b∈ Aand ∆(h) =^Ph₍₁₎⊗h₍₂₎ (Sweedler’s notation for the coproduct). If◦is another action of H onA, we say that the actions areisomorphic if there is an algebra automorphism φ:A → Asuch thath◦φ(a) =φ(h∗a) for anyh ∈H and any a∈ A. The actions are calledequivalent if there is an

automorphism of algebrasφ:A → Aand an automorphism of Hopf algebrasα:H→H such that α(h)◦φ(a) =φ(h∗a) for any h∈H and anya∈ A.

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IfH=FG then ∆(g) =g⊗g for g ∈G, so we have

g∗(ab) = (g∗a)(g∗b), hence the elements of a group act by automorphisms ofA. In an arbitrary Hopf algebraH the elements g with ∆(g) =g⊗g form a multiplicative subgroupG(H) whose elements are calledgrouplikes.The grouplikes act like the

automorphisms. The elementsx with ∆(x) =x⊗1 + 1⊗x are calledprimitive, they form a Lie subalgebraP(H) with respect to the commutator [x,y] =xy −yx. Primitive elements act like derivationsx∗(ab) = (x∗a)b+a(x∗b). Given grouplikesg,h, an elementx is called (g,h)-primitive if ∆(x) =x⊗g+h⊗x.

These element act like skew-derivations

x∗(ab) = (x∗a)(g∗b) + (h∗a)(x∗b).

We say that the action isinner if there is a convolution invertible linear mapu:H → Asuch that, for anyh∈H anda∈ Awe have

h∗a=^X

(h)

u(h₍₁₎)au⁻¹(h₍₂₎).

The convolution productfg off,g ∈Hom(H,A) is given by (fg)(h) =^Pf(h₍₁₎)g(h₍₂₎). An important result by A. Masuoka is the following.

Theorem

Any action of a Hopf algebra on a central simple algebra is inner.

This theorem generalizes Skolem–Noether’s Theorem saying that the automorphisms of matrix algebras are inner. The same is known for the (skew) derivations.

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In the case of a group algebraH=FG, we haveu⁻¹(g) =u(g)⁻¹ andg∗a=u(g)au(g)⁻¹, for all g ∈G. In the case of an

enveloping algebraH=U(g),u⁻¹(x) =−u(x) and x∗a=u(x)a−au(x) = [u(x),a], for allx ∈g. For a

(1,g)-primitive element x, that is ∆(x) =x⊗1 +g⊗x, then u⁻¹(x) =−u(g)⁻¹u(x) and x∗a=u(x)−(u(g)au(g)⁻¹)u(x).

In what follows, we will talk about the actions of finite-dimensional Hopf algebras with an abelian group of grouplikesG(H), which are generated byG(H) and some skew-primitive elements. It is known [2] that these are exactlypointed Hopf algebras with an abelian group of grouplikes.

LetG=G(H) be the group of group-like elements in H. Denote byGthe group of multiplicative F-characters ofG. From now on we will be assuming thatG is abelian andF is such that|G|=|G|

(for instance,F algebraically closed of characteristic coprime to

|G|, or F=Rand|G|= 2, etc.). In this case, it is well-known that the algebraAacquires a G-grading

Γ :A=⊕_φ∈GA_φ (1)

given by

A_φ ={a∈ A |g ∗a=φ(g)a for anyg ∈ G}.

Ifa∈ A_φ we also write dega=φ. The set SuppΓ ={φ∈G| A_φ ̸={0}}

is called thesupport of the grading Γ. The subspacesA_γ are called homogeneous components ofA. An elementa∈ A_γ is called homogeneous of degree γ. One writes dega=γ. A subspaceB is calledgraded if B=^L_γ∈G(B ∩ A_γ).

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Group gradings onMn(F) have been completely described over arbitrary algebraically closed fieldsF([6], see also [17], [9, Chapter 2] and references therein). Let us call a graded algebraA

graded-simple if it has no nontrivial proper graded ideals. A G-graded algebra D is called graded-divisionif all nonzero

homogeneous elements ofD are invertible. In full analogy with the classical Wedderburn - Artin theorem, the following is true.

Theorem

LetGbe a group and letA be a G-graded algebra over a fieldF.

IfAis graded simple satisfying the descending chain condition on graded left ideals, then there exists aG-graded algebra D and a G-graded right D-module V such thatDis a G-graded division algebra, V is finite-dimensional overD, andA is isomorphic to End_DV as a G-graded algebra.

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Note that if{e₁, . . . ,e_m} is a graded basis ofV as a right vector space overD, with degei =γi, i = 1, . . . ,n, thenEndDV is spanned by the elementsEijd where d is a graded element of D andE_ij(e_j) =e_i. The grading is given by

degEijd =γi(degd)γ_j⁻¹.

Using Kronecker products, we can writeA ∼=Mn(F)⊗ D and identifyE_ijd with E_ij⊗d. Then deg(E_ij⊗d) =γ_i(degd)γ_j⁻¹.

IfD=F, the above grading is called elementary, defined by the n-tuple γ₁, . . . , γ_n. If dimDV = 1 then the grading is called division. In this latter case, any nonzero a∈ A_χ is invertible, for anyχ∈G. We will use the termselementary anddivision also for the actions of a Hopf algebraH, with G(H) abelian, on a matrix algebraMm(F) in the case where the respective grading of Mm(F) byGis elementary or division.

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If the grading is elementary and dim_FV =n then the grading is completely defined by thedimension function κ: Γ→N0 defined as follows. LetV =⊕_γ∈ΓVγ. Then κ(γ) = dimVγ. The groupG naturally acts on the set of such functions if one defines

(γ∗κ)(γ^′) =κ(γγ^′). Two elementary gradings, defined by the dimension functionsκ andκ^′, are isomorphic if and only if κ^′ ∈G∗κ. Let us take a graded basis{e₁, . . . ,en} inV, with dege_i =γ_i. Thenκ(γ) is the number of elements e_i in the basis such that degei =γ. A graded basis of End_FV is then formed by the mapsEij, as above, and degEij =γiγ⁻¹_j .

No description of graded division algebrasD over arbitrary fields is available, but in the case of an algebraically closed field of

characteristic zero or coprime to the orders of elements inG they are known to be isomorphic to the twisted group algebras

D=F^σT, whereσ ∈Z²(T,F^×) is a 2-cocycle (actually,

bicharacter) onT with values inF^×. Two algebras corresponding to the cocyclesσ₁ andσ₂ are graded isomorphic if and only if σ₁ andσ2 belong to the same cohomology class inZ²(T,F^×). In the cohomology class of a bicharacterσ there is precisely one

alternating bicharacter β:T×T→F^×, given by β(φ, ψ) = σ(φ, ψ)

σ(ψ, φ).

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An algebraD=F^σTis simple if and only if β:T×T→F^× is nonsingular, that is,Kerβ={φ|β(φ, ψ) = 1, for all ψ∈T} is trivial. In the Theorem 2, in the case of simple algebras, the graded division algebraDis simple, as an ungraded algebra. As a result, D is defined, up to isomorphism, by its supportTand a nonsingular alternating bicharacterβ:T×T→F^×. Different bicharacters lead to nonisomorphic graded division algebrasD(T, β). Note that each homogeneous componentD_τ of D(T, β) is one dimensional, and its basis{X_τ|τ ∈T}, can be so chosen that

X_τ^o(τ) =I and Xτ2Xτ1 =β(τ2, τ1)Xτ1Xτ2, o(τ) the order of τ ∈T.

(2)

Theorem

([4], also [9, Chapter 2]) LetTbe a finite abelian group and let F be an algebraically closed field. There exists a grading on the matrix algebra Mn(F) with support Tmaking Mn(F) a graded division algebra overTif and only if char F does not divide n and

T∼=Z²ℓ1× · · · ×Z²ℓr, whereℓ1· · ·ℓr =n. (3) The isomorphism classes of such gradings are in one-to-one

correspondence with nondegenerate alternating bicharacters β:T×T→F^×. All such gradings belong to one equivalence class.

There is a well known realization of the division grading onMn(F) in terms of Sylvester’sclock andshift matrices.

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Now we can make conclusions about theaction of the group algebra H=FG of an abelian groupG on a matrix algebra A.

Let, as before,Gbe the group of characters of G=G(H).

According to [3] (see also [9, Corollary 2.43], if we are given a G-grading of A=M_n(F)∼=End_D(VD), then the matrixu(g) of the action of for anyg ∈ G, in an appropriate basis ofV_D, can be written as follows. LetT⊂Gbe the support ofD so that

D_τ =Span X_τ, for eachτ ∈T. Suppose that σ₁, . . . , σ_s are some elements ofGwhich are pairwise non-congruentmodTand such that

VD = (V_σ1⊗ D)⊕ · · · ⊕(V_σs⊗ D)

is the decomposition ofV as a graded vector space over D, and

Then the action ofg ∈ G can be given as the conjugation by the matrix

u(g) = (σ₁(g)I_d1⊗X_f(g))⊕ · · · ⊕(σ_s(h)I_ds⊗X_f(g)), (4) wheref(g) is a uniquely defined element ofT such that

τ(g)Xτ =g∗Xτ =X_f(g)XτX_f⁻¹_(g)=β(f(g), τ)Xτ. We haveβ(f(g), τ) =τ(g). Since β is nonsingular, one easily checks that the mapf :g 7→f(g) is a well-defined homomorphism fromG toT.

The isomorphism classes of these actions are described by the same parameters as the isomorphism classes of the corresponding gradings.

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IfH is a pointed non-cosemisimple finite-dimensional Hopf algebra with an abelian groupG of group-likes then there are always elementsh,k ∈ G, a character χ:G →F^× with χ(g)̸= 1 and an elementx ̸∈FG such that

gxg⁻¹ =χ(g)x for all g ∈ G, (5)

∆(x) =x⊗h+k⊗x. (6)

For an (h,k)-primitive elementx, one has

ε(x) = 0 andS(x) =−k⁻¹xh⁻¹.

If a Hopf algebra is generated byG andx, as above, one can replacex by a (g,1) primitive element or (1,g)-primitive element,

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 gather few simple properties.

Proposition

LetΓbe a grading defined in (1). Suppose x is an element of H such that there is a characterχ:G →F^× satisfying hx =χ(h)xh, for all h∈ G. LetK be the kernel of the action of G onA,K^⊥ its orthogonal complement inG,T the subgroup generated bySuppΓ inG. Then the following are true.

(i) The action of x on A maps anyA_γ to A_χγ. (ii) Ifχ̸∈ K^⊥ then x acts trivially on A.

(iii) Ifχ̸∈T then x acts trivially on A.

(iv) If there is natural m such that (SuppΓ)^m ={ε}and χ^m̸=ε then x acts trivially on A.

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Now letH contain a group-like element g and a (1,g)-primitive elementx such that

gⁿ= 1,xⁿ= 0, gx =ωxg, whereω is a primitiventh root of 1.

(7) We have

∆(g) =g⊗g, ∆(x) =x⊗1 +g⊗x, ε(g) = 1, ε(x) = 0, S(g) =g⁻¹, S(x) =−g⁻¹x.

One calls such a pair of elements aTaftω-pair. In this case, g^sx =χ(g^s)g^sx, whereχ(g^s) =ω^s. Let G be the cyclic subgroup generated byg.

Ann²-dimensional Hopf algebra generated by an ω-pair g,x is

Since in this caseG∼=Zn, it follows from the above, that A acquires aZn-grading, as follows. Letχ∈Gbe given by χ(g) =ω. SetA_k =A_χk. Then A=⊕ⁿ⁻¹_k=0A_k is the desired grading.

Corollary

If the action ofG is not faithful then the action of x is trivial. If the support of the grading is contained in a proper subgroupTof Gthen the action of x is trivial.

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An algebraA with identity element 1 is calledcentral if its center coincides withF.1. Suppose that a Hopf algebra H acts onA, and this action is inner via the convolution invertible mapu:H → A.

Then, for any group-like elementg ∈ G and any (h,k)-primitive elementx,h,k ∈ G, one has

g∗a=u(g)au(g)⁻¹, x∗a=u(x)au(h)⁻¹−u(k)au(k)⁻¹u(x)u(h)⁻¹. (8) In the case wherex is (1,k)-primitive, we would have

g∗a=u(g)au(g)⁻¹, x∗a=u(x)a−u(k)au(k)⁻¹u(x). (9) In the case wherex is (h,1)-primitive, we would have

g∗a=u(g)au(g)⁻¹, x∗a=u(x)au(h)⁻¹−au(x)u(h)⁻¹= [u(x),a]u(h)⁻¹. (10)

Here [a,b] =ab−ba is the usual commutator ofa,b∈ A.

Theorem

Let H be a finite-dimensional pointed Hopf algebra with abelian group of group-likes,Aa matrix algebra. If two inner actions ∗ and◦defined by the functions u :H→ A and v :H→ A are isomorphic then there exists an invertible element C ∈ Aand λ(g), µ(g,x)∈F, such that u(g) =λ(g)Cv(g)C⁻¹, where λ(g)∈F, for any grouplike g ∈H and

u(x) =Cv(x)C⁻¹+µ(g,x)u(g),µ(g,x)∈F, for any

(1,g)-primitive element x ∈H. In the case of a (g,1)-primitive element y , we must have u(y) =Cv(y)C⁻¹+µ(g,y)I.

Conversely, suppose our conditions hold andH is generated by its group-likes and skew-primitives. Then

v(g)⁻¹C⁻¹u(g)C⁻¹ and v(g)⁻¹C⁻¹u(x)C −v(g)⁻¹v(x) are central elements. We can read our calculations in the reverse order, to find thatφ(A) =CAC⁻¹ preserves the action of group-like and skew-primitive elements. SinceH is generated by these, we find thatφis an isomorphism of actions ∗and◦.

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Pointed Hopf algebras with abelian group of grouplikes have been classified thanks to the efforts of many authors. An incomplete list of names include D. Radford, N. Andruskiewitchsh, H.-J.

Schneider, I. Heckenberger, I. Angiono,... A useful survey of the development in the area is [2]. We looking here at a special case of actions of a class of pointed Hopf algebras, as appears in the paper [1] by N. Andruskiewitchsh and H.-J. Schneider.

LetG be a finite nontrivial abelian group. A datumfor aquantum linear space R=R(a₁, . . . ,an;χ1, . . . , χn) consists of the

elementsa₁, . . . ,a_n∈ G,χ₁, . . . , χ_n∈G=G, such that the^b following conditions are satisfied:

q_j =χ_j(a_j)̸= 1 (11) χj(ai)χi(aj) = 1 for all 1≤i ̸=j ≤n. (12) One says that the datum, or its associated quantum linear space, has rankn. IfN_j =o(q_j), the order ofq_i in G, then

R=R(a₁, . . . ,a_n;χ₁, . . . , χ_n) is an algebra with the following presentation in terms of generators and defining relations:

R=⟨x₁, . . . ,x_n|x_i^Ni = 0; x_ix_j =χ_j(a_i)x_jx_i for all 1≤i ̸=j ≤n⟩.

(13)

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Let us fix a decompositionG= (g1)× · · · ×(gk) and denote byMℓ

the order ofg_ℓ, 1≤ℓ≤k. Let a1, . . . ,an∈ G,χ1, . . . , χn∈G be the datum for a quantum linear space

R=R(a₁, . . . ,an;χ1, . . . , χn), that is, (11, 12) hold.

A compatible datumD for G andRconsists of a vector

(µ₁, . . . , µ_n),µ_i ∈ {0,1} and a matrix (χ_ij),λ_ij ∈F, 1≤i,j ≤n, such that

(i) if a_i^Ni = 1 or χ^N_i ⁱ ̸=ε, thenµ_i = 0, (ii) if a_ia_j = 1 or χ_iχ_j ̸=ε, then λ_ij = 0.

Definition

LetG be a finite abelian group,R a quantum linear space, andD a compatible datum. Keep the notation above. LetP(G,R,D) be the algebra presented by generatorsg_ℓ, 1≤ℓ≤k , andx_i,

1≤i≤n, with defining relations

g_ℓ^Mℓ = 1,1≤ℓ≤k; (14) g_ℓgt=gtg_ℓ, 1≤ℓ,t ≤k; (15) xigℓ=χi(gℓ)gℓxi, , 1≤ℓ≤k; (16) x_i^Ni =µi(1−a^N_i ⁱ),1≤i ≤n; (17) x_jx_i =χ_i(a_j)x_ix_j +λ_ij(1−a_ia_j),1≤i,j ≤n. (18) A unique Hopf algebra structure onP(G,R,D) is given by

∆(g_ℓ) =g_ℓ⊗g_ℓ, ∆(xi) =xi⊗1 +ai⊗xi,1≤ℓ≤k, 1≤i,j ≤n;

S(gℓ) =g_ℓ⁻¹, S(xi) =−a_i⁻¹xi, 1≤ℓ≤k, 1≤i ≤n.

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One can also say thatH is generated by its groupG =G(H) of group-likes, which is abelian, and the set of (1,a_i)-primitivex_i, for some elementa_i ∈ G,i = 1, . . . ,n. These elements satisfy the above relations (16, 17, 18). In these relations,χ1, . . . , χn are multiplicative charactersχ_i :G →F^×,N_i =o(χ_i(a_i))̸= 1. If one denotesq_i =χ_i(a_i), then N_i is the order ofq_i in the groupF^×. The elementsµi andλij are subject to the compatibility conditions (i), (ii), just before Definition 8.

Finally, the numbern of generators of the quantum vector spaceR is called therank of the Hopf algebra P(G,R,D).

Remark

LetP(G,R,D) act on an algebraAso that, for some i = 1, . . . ,n, the action of x is nonzero. If g acts trivially on Athen χ(g) = 1.

We setH =P(G,R,D), as above.

Theorem

Let u:H→ A define an inner action of a Hopf algebra H on a matrix algebraA. Then for each i = 1, . . . ,k, there is a function λ_i :G →Fsuch that, for any g ∈ G one has

u(g)u(xi)u(g)⁻¹ =χi(g)u(xi) +λi(g)u(ai). (19) One can choose u so thatλ(g) = 0, that is,

u(g)u(x_i) =χ_i(g)u(x_i)u(g), for all g ∈ G, (20) the action remaining the same.

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An important consequence of this theorem is the following.

Corollary

Let a Hopf algebra H with a group of group-likesG andG=G^bact on a matrix algebraA. Let

Γ :A= ^M

φ∈G

A_φ

be the associatedGgrading of A. If x is a(1,a)-primitive element in H for some a∈ G andχ∈Gis such that gx =χ(g)x , for all g ∈ G. Then u(x)∈ A_χ.

LetH=P(G,R,D), as above.

Theorem

Let u:H→ A define an inner action of a Hopf algebra H on a matrix algebraA. Then there isσi ∈F such that the following relation holds for the elements u(a_i) and u(x_i)of the algebra A:

u(xi)^Ni =µi.1A+σiu(ai)^Ni. (21)

Theorem

The following relations hold for the elements of the inner action of H=P(G,R,D) on a matrix algebraA, defined by the convolution invertible map u:H→ A . For any1≤i,j ≤n there existζij ∈F such that

u(xj)u(xi)−χj(ai)u(xi)u(xj) =λij.1A+ζiju(ai)u(aj). (22)

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Theorems 10, 12 and 13 work for Taft algebras and their doubles, but also Lusztig’s kernel and so on. Not only do they give the necessary but also they give sufficient conditions for the actions of H on matrix algebras. Although it seems impossible to classify all sets of matrices satisfying all the relations given in these theorems, we provide some interesting examples.

The action of the Taft algebraTn(ω) on a simple algebra A was treated in a paper by A. Gordienko [11], in connection with the codimension growth of identities of finite-dimensional algebras with the action of Hopf algebras. Gordienko proved that one can choose u(g) andu(x) so thatu(g)u(x)u(g)⁻¹ =ω⁻¹u(x),u(g)ⁿ=I and u(x)ⁿ=λI_n.

Next we talk about examples. First we consider Hopf algebras of rank 1.

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Theorem

Let a Hopf algebra H =P(G,R,D) with G= (g),R=F[x], be defined by g^m = 1, gx =χ(g)xg and xⁿ=µ(1−aⁿ). Suppose H acts on a matrix algebraA=End V . Then the action is inner, via u:H → A. There is an mth primitive rootζ of 1 such that V =^Pm−1_i=0 Vi, where Vi is an eigenspace for u(a), with eigenvalue ζⁱ. If m=nr then V =^Pr_t=0W_t, each W_t being invariant under both u(g) and u(x). In an appropriate basis of W_t, the matrix of u(x) has the form of (23), while the matrix of u(a)is scalar with coefficientζ^t+kr. If H is nilpotent, then either u(x)ⁿ= 0 or u(x) takes the form (24). 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 takes

u(x)^(t)=













0 0 . . . 0 u_1ℓ^(t) u^(t)₂₁ 0 . . . 0 0

. . . . . . . . . . . . . . . 0 0 . . . u_ℓ1^(t) 0













. (23)

where eachu^(t)_i+1,i is a rectangulard_k+1^(t) ×d_k^(t) matrix of rankr_i^(t). Here dimV_t+kr =d_k^(t) andr_i^(t)=rank u^(t)_i+1,i.

u(x)^(t) =











0 0 . . . 0 λI_dt I_dt 0 . . . 0 0 . . . . . . . . . . . . . . .

0 0 . . . I_dt 0











(24)

u(x)^(t) =











0 0 . . . 0 µtI_dt I_dt 0 . . . 0 0 . . . . . . . . . . . . . . .

0 0 . . . I_dt 0











. (25)

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Theorem

Let H=P(G,R,D)be a pointed Hopf algebra of rank 1, with admissible data D= (a, χ, µ), o(χ(a)) =n. LetGbe the group of characters onG. Assume that the action ofG on a matrix algebra AmakesA aG-graded division algebra, with supportT. In any extension of this action ofG to H we must have u(x) =αX_χ, for someα∈F. If χ̸∈T then the action of x is zero. Ifχ∈Tand χⁿ=f(a)ⁿthen extensions always exist, with any α∈F. If µ= 0, this condition is also necessary. Ifµ= 1, in which caseχⁿ=εand aⁿ̸=e, the extensions exist even if aⁿ acts non-trivially. In this case,αⁿ=−1.

The actions via u(x) =αX_χ and u^′(x) =α^′X_χ are isomorphic if

The Drinfeld double of a Taft algebra is a particular case of algebrasP(G,R,D) defined earlier. We will use the following presentation of the Drinfeld doubleH=D(Tn) of the Taft algebra Tn(ω), in the case where n is an odd number >1. As an algebra, H is generated byg,x,G,X, where g,G are group-likes and x,X are skew-primitives and

gⁿ=Gⁿ= 1, xⁿ=Xⁿ= 0,

gG =Gg, gx =ω⁻¹xg, Gx =ω⁻¹xG, gX =ωXg, GX =ωXG, xX −ωXx = 1−gG

∆(x) =x⊗1 +g⊗x, ∆(X) =X⊗1 +G⊗X.

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Because the group of grouplikes in the case ofD(Tn) is isomorphic toZn×Zn, there are division actions of D(T_n) on some matrix algebras. Possible actions in the casen = 3 are the following.

Case 1: u(g) =







0 1 0 0 0 1 1 0 0





, u(G) =







1 0 0

0 ω² 0

0 0 ω





,

u(x) =γ







0 1 0

0 0 ω²

ω 0 0





, u(X) =δ







0 0 ω²

1 0 0

0 ω 0





.

Case 2: u(g) =







0 0 1 1 0 0 0 1 0





, u(G) =







1 0 0

0 ω 0

0 0 ω²





,

We adopt the following presentation ofH=uq(sl2) from [1]. H is generated by the elementsa,x,y such that

aⁿ = 1, xⁿ=yⁿ= 0

ax = ω²xa, ay =ω⁻²ya, xy−yx =a−a⁻¹. (26) Heren is an odd number,n≥3.

The coproducts are given by

∆x =x⊗a+ 1⊗x, ∆y =y⊗1 +a⁻¹⊗y,∆a=a⊗a.

Hereω is a primitiventh root of 1, n an odd number.

SupposeH acts on a matrix algebra A. Then the action is inner, via a convolution invertible mapu :H→ A and given by the formulas below, whereAis any element of A.

a∗A = u(a)Au(a)⁻¹

x∗A = u(x)Au(a)⁻¹−Au(x)u(a)⁻¹

y∗A = u(y)A−u(a)⁻¹Au(a)u(y) (27)

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Proposition

The elements u(a),u(x),u(y)∈ Aon the right hand sides of the equations in (27) can be so chosen that

u(a)ⁿ=θ1, (28)

u(a)u(x) =ω²u(x)u(a), (29) u(a)u(y) =ω⁻²u(y)u(a), (30) u(x)ⁿ=µ1, u(y)ⁿ=ν1 (31) u(x)u(y)−u(y)u(x) =u(a)−τu(a)⁻¹, (32)

It is well-known thatu_q(sl₂) is a Hopf homomorphic image of D(H) , for H =T_n2(ω) , by settingω =q² and defining Φ : D(H)→uq as follows: G7→K ,g 7→K⁻¹ ,x 7→F , and X 7→E K⁻¹ :=−(q−q⁻¹)E.

MoreoverKer(Φ) =D(H)kG⁺, whereG is the cyclic subgroup generated bygG.

Corollary

Each action of u_q(sl₂)on an algebraA lifts to an action of the Drinfeld double D(Tq)onA via the homomorphism shown above.

Thus any action of u_q(sl₂) on M_n(F)can be obtained from an elementary action of D(T_q).

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Note that the converse is not true: no division action of the Drinfeld doubleD(T_q) comes as a lifting of an action of the u_q(sl₂), becauseG andg do not act as inverses of each other.

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