Hopf algebra automorphisms

, 33: 3029–3050, 2005 Copyright © Taylor & Francis, Inc.

ISSN: 0092-7872 print/1532-4125 online DOI: 10.1081/AGB-200066110

GRADED QUANTUM GROUPS AND QUASITRIANGULAR HOPF GROUP-COALGEBRAS^#

Alexis Virelizier

Max Planck Institute for Mathematics, Bonn, Germany

Starting from a Hopf algebra endowed with an action of a group by Hopf automorphisms, we construct by a “twisted” double methoda quasitriangular Hopf -coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf -coalgebras for any finite groupand for infinite groups such as GLnkkk. In particular, we define the graded quantum groups, which are Hopf-coalgebras for =h^land generalize the Drinfeld-Jimbo quantum enveloping algebras.

Key Words: Drinfeld double; Graded quantum groups; Hopf algebra automorphisms; Quasi- triangular Hopf group-coalgebras.

2000 Mathematics Subject Classification: 81R50; 17B37; 16W30.

INTRODUCTION

Letbe a group. Turaev (2000) introduced the notion of abraidedcategory and showed that such a category gives rise to a 3-dimensional homotopy quantum field theory (the target being aK1space). Moreover braided-categories, also called -equivariant categories, provide a suitable mathematical formalism for the description of orbifold models that arise in the study of conformal field theories in whichis the group of automorphisms of the vertex operator algebra, see Kirillov (2004).

The algebraic structure whose category of representations is a braided - category is that of a quasitriangular Hopf-coalgebra, see Turaev (2000), Virelizier (2002). The aim of the present article is to construct examples of quasitriangular Hopf -coalgebras. Note that quasitriangular Hopf -coalgebras are also used in Virelizier (2001) to construct HKR-type invariants of flat -bundles over link complements and over 3-manifolds.

Following Turaev (2000), a Hopf -coalgebra is a family H =H_∈ of algebras (over a field k) endowed with a comultiplication = H → H⊗H _∈, a counit H₁→k, and an antipodeS=S H→H−1_∈ which verify some compatibility conditions. A crossing for H is a family of algebra

Received March 15, 2004; Revised January 15, 2005; Accepted March 6, 2005

#Communicated by H. J. Schneider.

Address correspondence to Alexis Virelizier, Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany; Fax: +49-228-402277; E-mail: [email protected]

3029

isomorphisms= H→H −1 _∈, which preserves the comultiplication and the counit, and which yields an action ofin the sense that = . A crossed Hopf -coalgebra H is quasitriangular when it is endowed with an R-matrix R=R ∈H⊗H _∈ verifying some axioms (involving the crossing ) which generalize the classical ones given in Drinfeld (1987). Note that the case=1 is the standard setting of Hopf algebras.

Starting from a crossed Hopf -coalgebra H=H_∈, Zunino (2004) constructed a double ZH=ZH_∈ of H, which is a quasitriangular Hopf - coalgebra in which H is embedded. One has that ZH=H⊗ _∈H^∗

as a vector space. Unfortunately, each component ZH is infinite-dimensional (unless H =0 for all but a finite number of ∈).

To obtain non-trivial examples of quasitriangular Hopf -coalgebras with finite-dimensional components, we restrict ourselves to a less general situation: our initial datum is not any crossed Hopf-coalgebra but a Hopf algebra endowed with an action ofby Hopf algebra automorphisms. Remark indeed that the component H₁ of a Hopf-coalgebraH =H_∈ is a Hopf algebra and that a crossing for H induces an action ofon H₁by Hopf automorphisms.

In this article, starting from a Hopf algebraAendowed with an action → Aut_HopfA of a group by Hopf automorphisms, we construct a quasitriangular Hopf-coalgebraDA =DA _∈. The algebraDA is constructed in a manner similar to the Drinfeld double (in particularDA =A⊗A^∗as a vector space) except that its product is “twisted” by the Hopf automorphism A→A.

The algebraDAid_Ais the usual Drinfeld double. Note that the algebrasDA andDA are in general not isomorphic when = .

This method allows us to define non-trivial examples of quasitriangular Hopf -coalgebras for any finite group and for infinite groups such as GL_nk. In particular, given a complex simple Lie algebraof rankl, we define the graded quantum groups U_h∈∗^l and U_h∈hl, which are crossed Hopf group- coalgebras. They are obtained as quotients of DU_q+ and DU_hb₊ , where ₊ denotes the Borel subalgebra of , is an action of ^∗l by Hopf automorphisms of U_q+, and is an action ofh^l by Hopf automorphisms ofU_h+. Furthermore, the crossed Hopfh^l-coalgebraU_h∈hl is quasi- triangular.

The article is organized as follows. In Section 1, we review the basic definitions and properties of Hopf -coalgebras. In Section 2, we define the twisted double of a Hopf algebraA endowed with an action of a group by Hopf automorphisms.

In Section 3, we explore the caseA=kG, whereGis a finite group. In Section 4, we give an example of a quasitriangular Hopf GL_nk-coalgebra. Finally, we define the graded quantum groups in Sections 5 and 6.

Throughout this article,is a group (with neutral element 1) andk is a field.

Unless otherwise specified, the tensor product⊗ =

k is assumed to be overk. 1. HOPF GROUP-COALGEBRAS

In this section, we review some definitions and properties concerning Hopf group-coalgebras. For a detailed treatment of the theory of Hopf group-coalgebras, we refer to Virelizier (2002).

1.1. Hopf-Coalgebras

AHopf -coalgebra(over k) is a family H =H_∈ of k-algebras endowed with a family = H →H⊗H _∈ of algebra homomorphisms (the comultiplication) and an algebra homomorphism H₁→k (thecounit) such that, for all ∈,

⊗id_H

= id_H

⊗

(1.1)

id_H⊗

₁=id_H =

⊗id_H

₁ (1.2)

and with a family S=S H→H−1_∈ of k-linear maps (the antipode) which verifies that, for all∈,

mS−1⊗id_H−1=1=mid_H⊗S−1−1 (1.3) wherem H⊗H→H and 1∈H denote, respectively, the multiplication and unit element ofH.

When=1, one recovers the usual notion of a Hopf algebra. In particular H₁ m₁1_{1 11} S₁is a Hopf algebra.

Remark that the notion of a Hopf -coalgebra is not self-dual and that if H=H_∈ is a Hopf-coalgebra, then∈H=0is a subgroup of.

A Hopf -coalgebra H=H_∈ is said to be of finite type if, for all ∈, H is finite-dimensional (overk). Note that it does not mean that

∈H is finite- dimensional (unlessH=0 for all but a finite number of∈).

The antipode of a Hopf-coalgebra H=H_∈ is anti-multiplicative: each S H→H−1 is an anti-homomorphism of algebras, and anti-comultiplicative:

S₁=and −1⁻¹S =_{H−1H −1}S⊗S for any ∈, see Virelizier (2002, Lemma 1.1).

The antipode S=S_∈ of H=H_∈ is said to be bijective if each S is bijective. As for Hopf algebras, the antipode of a finite type Hopf -coalgebra is always bijective, see Virelizier (2002, Corollary 3.7(a))).

1.2. Crossed Hopf-Coalgebras

A Hopf-coalgebraH=H_∈ is said to becrossedif it is endowed with a family=

H→H −1

∈ of algebra isomorphisms (thecrossing) such that, for all ∈,

⊗ = −1 ⁻¹ (1.4)

= (1.5)

= (1.6)

It is easy to check that₁H =id_H and S=S −1 for all ∈.

1.3. Quasitriangular Hopf-Coalgebras

A crossed Hopf -coalgebra H=H_∈ is said to be quasitriangular if it is endowed with a family R=R ∈H⊗H _∈ of invertible elements (the R-matrix) such that, for all ∈andx∈H ,

R · x= −1⊗id_H −1x·R (1.7)

id_H⊗ R =R_{1 3}·R ₁₂ (1.8)

⊗id_HR =id_H⊗ −1R −1_{1 3}·R ₂₃ (1.9)

⊗ R=R −1 ⁻¹ (1.10)

where denotes the flip mapH ⊗H→H⊗H and, fork-spacesP Qandr =

jp_j⊗q_j∈P⊗Q, we set r₁₂=r⊗1∈P⊗Q⊗H, r₂₃=1⊗r ∈H⊗P⊗Q, andr_{1 3}=

jp_j⊗1 ⊗q_j∈P⊗H ⊗Q.

Note thatR₁₁ is a (classical)R-matrix for the Hopf algebraH₁.

When is abelian and is trivial(that is, H =id_H for all ∈), one recovers the definition of a quasitriangular-colored Hopf algebra given in Ohtsuki (1993).

The R-matrix always verifies (see Virelizier, 2002, Lemma 6.4) that, for any ∈,

⊗id_HR₁=1=id_H⊗R₁ (1.11) S−1⊗id_H R−1 =R⁻¹ and id_H⊗S R⁻¹=R −1 (1.12) S⊗S R =⊗id_{H −1}R−1 −1 (1.13) and provides a solution of the-colored Yang-Baxter equation:

R ₂₃·R_{1 3}·R ₁₂=R ₁₂·id_H⊗ −1R −1_{1 3}·R ₂₃ (1.14)

1.4. Ribbon Hopf-Coalgebras

A quasitriangular Hopf -coalgebra H=H_∈ is said to be ribbon if it is endowed with a family=∈H_∈ of invertible elements (the twist) such that, for any ∈,

x=⁻¹x for all x∈H (1.15)

S=−1 (1.16)

= −1 (1.17)

=⊗ · −1⊗id_HR −1·R (1.18) Note that₁ is a (classical) twist of the quasitriangular Hopf algebraH₁.

1.5. Hopf-Coideals

Let H=H_∈ be a Hopf -coalgebra. A Hopf -coideal of H is a family I=I_∈, where eachI is an ideal ofH, such that, for any ∈,

I ⊂I⊗H +H⊗I (1.19)

I₁=0 (1.20)

SI⊂I−1 (1.21)

The quotient H=H=H/I_∈, endowed with the induced structure maps, is then a Hopf-coalgebra. IfH is furthermore crossed, with a crossing such that, for any ∈,

I⊂I −1 (1.22)

then so isH (for the induced crossing).

2. TWISTED DOUBLE OF HOPF ALGEBRAS

In this section, we give a method (the twisted double) for defining a quasitriangular Hopf-coalgebra from a Hopf algebra endowed with an action of a groupby Hopf automorphisms.

2.1. Hopf Pairings

Recall that aHopf pairingbetween two Hopf algebras A andB (overk) is a bilinear pairing A×B→k such that, for alla a∈Aandb b∈B,

a bb=a₁ ba₂ b (2.1)

aa b=a b₂a b₁ (2.2)

a1=a and 1 b=b (2.3)

Note that such a pairing always verifies that, for anya∈Aandb∈B,

Sa Sb=a b (2.4) since bothandS×Sare the inverse ofid×Sin the algebra Hom_kA×Bk endowed with the convolution product.

Let A×B→k be a Hopf pairing. Its annihilator ideals areI_A=a∈A a b=0 for allb∈B and I_B=b∈Ba b=0 for alla∈A. It is easy to check thatI_AandI_Bare Hopf ideals ofAandB, respectively. Recall thatis said to benon-degenerateifI_AandI_Bare both reduced to 0. A degenerate Hopf pairing A×B→k induces (by passing to the quotients) a Hopf pairing A/I¯ _A×B/I_B→ k, which is non-degenerate.

Most of Hopf algebras we shall consider in the sequel will be defined by generators and relations. The following provides us with a method of constructing Hopf pairings, see Van Daele (1993), Kassel et al. (1997).

Let A (resp. B) be a free algebra generated by elements a₁ a_p (resp.

b₁ b_q) over k. Suppose that A andB have Hopf algebra structures such that eacha_ifor 1≤i≤p(resp.b_jfor 1≤i≤q) is a linear combination of tensors a_r⊗a_s (resp.b_r⊗b_s). Givenpqscalars_ij ∈k with 1≤i≤pand 1≤j≤q, there is a unique Hopf pairing A×B→k such that a_i b_j=_ij.

Suppose now thatA(resp.B) is the algebra obtained as the quotient ofA(resp.

B) by the ideal generated by elementsr₁ r_m∈A(resp. s₁ s_n∈B). Suppose also that the Hopf algebra structure inA(resp.B) induces a Hopf algebra structure inA(resp.B). Then a Hopf pairing A×B→k induces a Hopf pairingA×B→ k if and only if r_i b_j=0 for all 1≤i≤m and 1≤j≤q, and a_i s_j=0 for all 1≤i≤pand 1≤j≤n.

2.2. The Twisted Double Construction

Definition-Lemma 2.1. Let A×B→k be a Hopf pairing between two Hopf algebras A and B. Let A→A be a Hopf algebra endomorphism of A. Set DA B =A⊗B as a k-space. Then DA B has a structure of an associative and unitary algebra given, for anya a∈Aandb b∈B, by

a⊗b·a⊗b=a₁ Sb₁a₃ b₃aa₂⊗b₂b (2.5)

1_DAB=1_A⊗1_B (2.6)

Moreover, the linear embeddings A →DA B and B →DA B defined bya→a⊗1_B and b→1_A⊗b, respectively, are algebra morphisms.

Remark 2.2. (a) Note thatDA B id_A is the underlying algebra of the usual quantum double of AandB(obtained by using the Hopf pairing).

(b) If and are different Hopf algebra endomorphisms of A, then the algebras DA B and DA B are not in general isomorphic, see Remark 4.2.

Proof. Let a a a ∈A and b b b∈B. Using the fact that is a Hopf pairing andis a Hopf algebra endomorphism, we have that

a⊗b·a⊗b

·a⊗b

=a₁ Sb₁a₃ b₅a₁ Sb₂b₁

×a₃ b₄b₃aa₂a₂⊗b₃b₂b

=a₁ Sb₁a₃ b₅a₁ Sb₁a₂ Sb₂

×a₄ b₄a₅ b₃aa₂a₃⊗b₃b₂b

and

a⊗b·

a⊗b·a⊗b

=a₁ Sb₁a₅ b₃a₁a₂ Sb₁

×a₃a₄ b₃aa₂a₃⊗b₂b₂ b

=a₁ Sb₁a₅ b₃a₁ Sb₁a₂ Sb₂

×a₃ b₅a₄ b₄aa₂a₃⊗b₃b₂b

Hence the product is associative. Moreover 1_A⊗1_B is the unit element since a⊗b·1⊗1=1 Sb₁1 b₃a⊗b₂

=Sb₁b₃a⊗b₂=a⊗b

and

1⊗1·a⊗b=a₁ S1a₃1a₂⊗b

=a₁a₃a₂⊗b=a⊗b

Finally, for anya a∈Aandb b∈B, we have that

a⊗1·a⊗1=a₁ S1a₃1aa₂⊗1

=a₁a₃aa₂⊗1

=aa⊗1 and

1⊗b·1⊗b=1 Sb₁1 b₃1⊗b₂b

=Sb₁b₃1⊗b₂b

=1⊗bb

ThereforeA →DA B andB →DA B are algebra morphisms.

In the sequel, the group of Hopf automorphisms of a Hopf algebraA will be denoted by Aut_HopfA.

Theorem 2.3. Let A×B→k be a Hopf pairing between two Hopf algebras A and B, and →Aut_HopfA be group homomorphism (that is, an action of on A by Hopf automorphisms). Then the family of algebras DA B = DA B _∈ (see Definition 2.1) has a structure of a Hopf -coalgebra given, for anya∈A,b∈B, and ∈, by:

a⊗b= a₁⊗b₁⊗a₂⊗b₂ (2.7)

a⊗b=_Aa_Bb (2.8)

Sa⊗b=a₁ b₁a₃ Sb₃Sa₂⊗Sb₂ (2.9) Proof. The coassociativity (1.1) follows directly from the coassociativity of the coproducts of A and B and the fact that = . Axiom (1.2) is a direct consequence of_A=_A. Since₁=id_AandDA B id_Ais underlying algebra

of the usual quantum double ofAandB, the counitis multiplicative. Let us verify that is multiplicative. Leta a∈Aandb b∈B. On one hand we have:

a⊗b·a⊗b

= a₁ Sb₁a₃ b₃ aa₂⊗b₂b

= a₁ Sb₁a₄ b₄ a₁a₂⊗b₂b₁⊗a₂a₃⊗b₃b₂

One the other hand,

a⊗b· a⊗b

= a₁⊗b₁⊗a₂⊗b₂· a₁⊗b₁ ⊗a₂⊗b₂

= a₁ Sb₁ a₃ b₃ a₄ Sb₄a₆ b₆

× a₁ a₂⊗b₂b₁⊗a₂a₅⊗b₅b₂

= a₁ Sb₁ a₃ b₃Sb₄a₅ b₆

× a₁a₂⊗b₂b₁⊗a₂a₄⊗b₅b₂

= a₁ Sb₁a₄ b₄ a₁a₂⊗b₂b₁⊗a₂a₃⊗b₃b₂

Let us verify the first equality of (1.3). Let a∈A, b∈B, and ∈. Denote the multiplication inDA B bym. We have

mS−1⊗id_DAB−1a⊗b

=a₁ b₁ a₃ Sb₅a₄ S²b₄

×a₆ Sb₂Sa₂a₅⊗Sb₃b₆

=a₁ b₁a₃ Sb₅S²b₄

×a₅ Sb₂Sa₂a₄⊗Sb₃b₆

=a₁ b₁a₄ Sb₂Sa₂a₃⊗Sb₃b₄

=a₁ b₁a₂ Sb₂1⊗1

=a b₁Sb₂1⊗1=ab1⊗1

The second equality of (1.3) can be verified similarly.

Let A×B→k be a Hopf pairing between two Hopf algebras A and B, and →Aut_HopfA be an action of on A by Hopf automorphisms. An action →Aut_HopfB of on Bby Hopf automorphisms is said to be - compatible if, for alla∈A,b∈Band ∈,

a b=a b (2.10)

Lemma 2.4. Let A×B→k be a Hopf pairing between two Hopf algebras A and B. Let →Aut_HopfA and →Aut_HopfB be two actions of by

Hopf automorphisms. Suppose that is -compatible. Then the Hopf -coalgebra DA B =DA B _∈ (see Theorem 2.3) admits a crossing given, for anya∈A,b∈Band ∈, by

a⊗b= a⊗ b (2.11)

Proof. Let ∈. We have that 1_A⊗1_B= 1_A⊗ 1_B=1_A⊗1_B and, for anya a∈Aandb b∈B,

a⊗b· a⊗b

= −1 a₁ S b₁ a₃ b₃

× a a₂⊗ b₂ b

= a₁ Sb₁ a₃ b₃ a a₂⊗ b₂ b

=a₁ Sb₁a₃ b₃ aa₂⊗ b₂b

= a⊗b·a⊗b

Moreover and are bijective and so is . Therefore DA B → DA B −1is an algebra isomorphism.

Finally, for anya∈A,b∈Band ∈, we have that:

−1 ⁻¹ a⊗b= −1 a₁⊗ b₁⊗ a₂⊗ b₂

= −1 a₁⊗ b₁⊗ a₂⊗ b₂

= a₁⊗ b₁⊗ a₂⊗ b₂

= ⊗ a⊗b

a⊗b= a b=ab=a⊗b and

a⊗b= a⊗ b= a⊗ b= a⊗b

Hencesatisfies Axioms (1.4), (1.5) and (1.6).

Corollary 2.5. Let A×B→k be a Hopf pairing and →Aut_HopfAbe an action ofonAby Hopf automorphisms. Suppose thatis non-degenerate and thatA (and soB) is finite dimensional. Then there exists a unique action^∗ →Aut_HopfB which is -compatible. It is characterized, for anya∈A,b∈Band ∈, by

a ^∗b= −1a b (2.12)

Consequently the Hopf -coalgebra DA B =DA B _∈ (see Theorem 2.3) is crossed with crossing defined by = ⊗^∗ for any ∈.

Proof. Let ∈. Since is non-degenerate and A and B are finite dimensional, the mapb∈B→· b∈A^∗ is a linear isomorphism, and so (2.12) does uniquely

define a linear map ^∗ B→B. Since is a Hopf pairing and −1 is a Hopf algebra isomorphism of A, the map ^∗ is a Hopf algebra isomorphism of B.

Moreover ^∗ is an action since ^∗₁=id_B (because ₁=id_A) and a ^∗ b= −1⁻¹a b= −1−1a b=−1a ^∗b=a ^∗∗bfor anya∈ A,b∈Band ∈. Finally (2.12) says exactly that^∗is -compatible.

Theorem 2.6. Let A×B→k be a Hopf pairing between two Hopf algebras A and B, and →Aut_HopfA be an action of on A by Hopf automorphisms.

Suppose that is non-degenerate and thatA(and soB) is finite dimensional. Then the crossed Hopf -coalgebra DA B =DA B _∈ (see Corollary 2.5) is quasitriangular withR-matrix given, for all ∈, by

R =

i

e_i⊗1_B⊗1_A⊗f_i (2.13)

wheree_ii andf_ii are basis ofAand B, respectively, such that e_i f_j=_ij. Remark 2.7. (a) The element

ie_i⊗1_B⊗1_A⊗f_i∈A⊗B⊗A⊗B is cano- nical, i.e., independent of the choices of the basise_ii ofAandf_ii ofBsuch that e_i f_j=_ij.

(b) Note that the hypothesis A is finite dimensional ensures that the sum

ie_i⊗1_B⊗1_A⊗f_ilies inA⊗B⊗A⊗B. More generally, assume thatAandB are graded Hopf algebras with finite dimensional homogeneous components and that is compatible with the gradings. Then the quotient Hopf algebras A/I_A and B/I_B are also graded and can be identified via with the duals of each other.

Suppose also that the action respects the grading so does the quotient ¯ → Aut_HopfA/I_A. In this case, there exists a unique action→Aut_HopfB/I_Bwhich is ¯ -compatible, where¯ A/I¯ _A×B/I_B→kis the induced Hopf pairing. Then the Hopf -coalgebra DA/I_A B/I_B¯ ¯ is quasitriangular by the same construction as in Theorem 2.6.

Proof. Fix basise_iofAandf_iofBsuch thate_i f_j=_ij (such basis always exist sinceis non-degenerate). Note thatx=

ix f_ie_iandy=

ie_i yf_ifor anyx∈Aandy∈B.

Recall that, since

ie_i⊗1_B⊗1_A⊗f_i is the R-matrix of the usual quantum doubleDA B id_A, we have

ij

Se_ie_j⊗f_if_j=1_A⊗1_B (2.14)

i

e_i⊗f_i1⊗f_i2=

ij

e_ie_j⊗f_j⊗f_i (2.15)

i

e_i1⊗e_i2⊗f_i=

ij

e_i⊗e_j⊗f_if_j (2.16)

Let ∈. From (2.14) and sinceA(resp.B) can be viewed as a subalgebra of DA B (resp. DA B ) via a→a⊗1_B (resp. b→1_A⊗b), we get

thatR is invertible inDA B ⊗DA B with inverse R⁻¹ =

i

Se_i⊗1_B⊗1_A⊗f_i Leta∈A,b∈Band ∈. For allx∈A, we have that:

id_A⊗B⊗A⊗x·R · a⊗b

=

i

a₂ Sf_i1a₄ f_i3x f_i2b₂e_i a₁⊗b₁⊗a₃

=

i

S⁻¹a₂ f_i1a₄ f_i3x₁ f_i2x₂ b₂e_i a₁⊗b₁⊗a₃

=

i

a₄x₁ S⁻¹a₂ f_ix₂ b₂e_i a₁⊗b₁⊗a₃

=x₂ b₂a₄x₁ S⁻¹a₂a₁⊗b₁⊗a₃

=x₂ b₂a₂x₁⊗b₁⊗a₁

and, sincex₁⊗x₂⊗x₃⊗x₄=

ix₂ f_ix₁⊗e_i1⊗e_i2⊗e_i3, id_A⊗B⊗A⊗x· −1⊗id_H −1a⊗b·R

=

i

e_i1 Sb₂ e_i3 b₄x ^∗₋1b₁f_ia₂e_i2⊗b₃⊗a₁

=

i

e_i1 Sb₂e_i3 b₄x₁ b₁x₂ f_ia₂e_i2⊗b₃⊗a₁

=x₂ Sb₂x₄ b₄x₁ b₁a₂x₃⊗b₃⊗a₁

=x₁ b₁Sb₂x₃ b₄a₂x₂⊗b₃⊗a₁

=x₂ b₂ a₂x₁⊗b₁⊗a₁

Hence, since thex·spanB^∗, Axiom (1.7) is satisfied.

Let us verify Axiom (1.10). Let ∈. Since ^∗ is -compatible, the basis e_ii of A and ^∗f_ii of B satisfy e_i ^∗e_j=e_i f_j=_ij. Therefore we get that:

⊗ R=

i

e_i⊗1_B⊗1_A⊗^∗f_j=R −1 ⁻¹

Finally, let us check Axioms (1.8) and (1.9). Let ∈. Using (2.15), we have:

id_DAB⊗ R =

i

e_i⊗1_B⊗1_A⊗f_i1⊗1_A⊗f_i2

=

ij

e_ie_j⊗1_B⊗1_A⊗f_j⊗1_A⊗f_i

=R_{1 3}·R ₁₂

Likewise, using (2.16) and (1.10), we have:

⊗id_DABR =

i

e_i1⊗1_B⊗e_i2⊗1_B⊗1_A⊗f_i

=

ij

e_i⊗1_B⊗e_j⊗1_B⊗1_A⊗f_if_j

= ⊗id_DABR −1 _{1 3}·R ₂₃

=id_DAB⊗ −1R −1_{1 3}·R ₂₃

This completes the proof of the quasitriangularity ofDA B . The next corollary is a direct consequence of Corollary 2.5 and Theorem 2.6.

Corollary 2.8. Let A be a finite-dimensional Hopf algebra and →Aut_HopfA be an action of on A by Hopf algebras automorphisms. Recall that the duality bracket A⊗A^∗ is a non-degenerate Hopf pairing between A and A^∗cop. Then DA A^∗copA⊗A^∗ is a quasitriangular Hopf-coalgebra.

Remark 2.9. The group of Hopf automorphisms of a finite-dimensional semisimple Hopf algebra A over a field of characteristic 0 is finite (see Radford, 1990). To obtain non-trivial examples of (quasitriangular) Hopf -coalgebras for an infinite group by using the twisted double method, one has to consider non-semisimple Hopf algebras (at least in characteristic 0).

2.3. Theh-Adic Case

In this subsection, we develop the h-adic variant of Hopf group-coalgebras.

A technical argument for the need of h-adic Hopf group-coalgebras is that they are necessary for a mathematically rigorous treatment of R-matrices for quantized enveloping algebras endowed with a group action.

Recall that if V is a vector space over h, the topology on V for which the sets hⁿV+vn∈ are a neighborhood base of v∈V is called the h-adic topology. If V andW are vector spaces over h, we shall denote byV ˆ⊗W the completion of the tensor product spaceV ⊗hW in theh-adic topology. LetV be a complex vector space. Then the setV hof all formal power seriesf =

n=0v_nhⁿ with coefficients v_n∈V is a vector space over h which is complete in the h-adic topology. Furthermore, V h ˆ⊗W h=V⊗W h for any complex vector spacesV andW.

An h-adic algebra is a vector space A over h, which is complete in the h-adic topology and endowed with a h-linear map m A ˆ⊗A→A and an element 1∈Asatisfyingmid_A ˆ⊗m=mm ˆ⊗id_Aandmaˆ⊗1=a=m1 ˆ⊗afor alla∈A.

By an h-adic Hopf -coalgebra, we shall mean a family H=H_∈ of h-adic algebras which is endowed withh-adic algebra homomorphisms H → H ˆ⊗H ( ∈) and A→h satisfying (1.1) and (1.2), and with Ch- linear maps S H→H−1∈) satisfying (1.3). In the previous axioms, one has to replace the algebraic tensor products⊗by theh-adic completions ˆ⊗.

The notions of crossed and quasitriangularh-adic Hopf-coalgebras can be defined similarly as in Sections 1.2 and 1.3.

The definitions of Section 2 and Theorem 2.3 carry over almost verbatim to h-adic Hopf algebras. The only modifications are that A ˆ⊗B→h is h-linear and that the algebra DA B , where is an h-adic Hopf endomorphism of A, is built over the completion A ˆ⊗B of A⊗B in the h-adic topology. The reasoning of the proof of Theorem 2.6 give the following h-adic version.

Theorem 2.10. Let A ˆ⊗B→h be an h-adic Hopf pairing between two h-adic Hopf algebras A and B, and →Aut_HopfA be an action of on A by h-adic Hopf automorphisms. Suppose that is non-degenerate and that e_ii and f_ii are basis of the vector spaces A and B, respectively, which are dual with respect to the form . If R =

ie_i⊗1_B⊗1_A⊗f_i belongs to the h-adic completion DA B ˆ⊗DA B , then R=R _∈ is a R-matrix of the crossed h-adic Hopf-coalgebraDA B =DA B _∈.

3. THE CASE OF ALGEBRAS OF FINITE GROUPS

Let G be a finite group. In this section, we describe Hopf G-coalgebras obtained by the twisted double method from the Hopf algebrakG.

Recall that the Hopf algebra structure of the (finite-dimensional) k-algebra kGofGis given byg=g⊗g,g=1 andSg=g⁻¹for allg∈G. The dual ofkGis the Hopf algebraFG=k^Gof functionsG→k. It has a basise_g G→ k_g∈G defined by e_gh=_gh where _gg=1 and _gh=0 if g=h. The structure maps of FG are given by e_ge_h =_ghe_g, 1_FG=

g∈Ge_g, e_g=

xy=ge_x⊗e_y, e_g=_g1, andSe_g=e_g−1 for anyg h∈G.

Set G→Aut_HopfkG defined by h=h⁻¹. It is a well-defined group homomorphism (since any ∈G is grouplike in kG). By Corollary 2.8, this datum leads to a quasitriangular Hopf G-coalgebra DkG FG^cop _kG×FG , which will be denoted byD_GG=DG_∈G.

Let us describeD_GGmore precisely. Let ∈G. Recall that DGis equal tokG⊗FGas ak-space. The unit element and product ofDGare given, for allg g h h∈G, by

1_DG=

g∈G

1⊗e_g and g⊗e_h·g⊗e_h=_g⁻¹h⁻¹ghgg⊗e_h

The structure maps ofD_GGare given, for any ∈Gandg h∈G, by g⊗e_h=

xy=h

g ⁻¹⊗e_y⊗g⊗e_x

g⊗e_h=_h1

Sg⊗e_h=g⁻¹⁻¹⊗e_g−1h⁻¹g⁻¹ g⊗e_h=g⁻¹⊗e_h−1

The crossed Hopf G-coalgebra D_GG is quasitriangular and furthermore ribbon withR-matrix and twist given, for any ∈G, by

R =

gh∈G

g⊗e_h⊗1⊗e_g and =

g∈G

⁻¹g⊗e_g

Note thatⁿ =

g∈G⁻ⁿgⁿ⊗e_gfor anyn∈.

4. EXAMPLE OF A QUASITRIANGULAR HOPF GL_nkkk-COALGEBRA

In this section,k is a field whose characteristic is not 2. Fix a positive integer n. We use a (finite dimensional) Hopf algebra whose group of automorphisms is known to be the group GL_nkof invertiblen×n-matrices with coefficients ink(see Radford, 1990) to derive an example of a quasitriangular Hopf GL_nk-coalgebra.

Definition-Proposition 4.1. For =_ij∈GL_nk, let _n be the -algebra generatedg,x₁ x_n,y₁ y_n, subject to the following relations:

g²=1 x²₁= · · · =x²_n=0 gx_i= −x_ig x_ix_j= −x_jx_i (4.1) y²₁= · · · =y²_n=0 gy_i= −y_ig y_iy_j= −y_jy_i (4.2) x_iy_j−y_jx_i=_ji−_ijg (4.3) where 1≤i j≤n. The family n =n_∈GLnk has a structure of a crossed Hopf GL_nk-coalgebra given, for any =_ij∈GL_nk, = ij∈GL_nk, and 1≤i≤n, by:

g=g⊗g g=1 Sg=g (4.4)

x_i=1⊗x_i+

n k=1

kix_k⊗g x_i=0 Sx_i=

n k=1

_kigx_k (4.5) y_i=y_i⊗1+g⊗y_i y_i=0 Sy_i= −gy_i (4.6)

g=g x_i= ⁿ

k=1

_kix_k y_i= ⁿ

k=1

˜

_iky_k (4.7)

where˜_ij=⁻¹. Moreovern is quasitriangular with R-matrix given, for any ∈ GL_nk, by:

R =1

2_S⊆nx_S⊗y_S+x_S⊗gy_S+gx_S⊗y_S−gx_S⊗gy_S

Heren=1 n,x_∅=1,y_∅=1, and, for a nonempty subsetSofn, we letx_S = x_i

1· · ·x_i

s and y_S=y_i

1· · ·y_i

s wherei₁<· · ·< i_s are the elements ofS.

Remark 4.2. Note that the algebras_nand_nare in general not isomorphic when ∈GL_nk are such that = . For example, we have that n¹n for any

∈GL_nkwith=1. This can be shown by remarking that:

_n/_nn¹_n/¹_n¹_n Indeed_n/_nn=0 sinceg= ¹

ji−ijx_iy_j−y_jx_i∈_nn(for some 1≤i,j≤n such that _ji=_ij) and so 1=g²∈_nn. Moreover, in ¹_n/¹_n¹_n, we have that x_k=x_kg²=0 (since x_kg=gx_k= −x_kg and so x_kg=0) and likewise y_k=0.

Hence¹n/¹n¹n=kgg²=1 0.

Proof. Let A_n be the k-algebra generated by g x₁ x_n, which satisfy the relations (4.1). The algebra A_n is 2ⁿ⁺¹-dimensional and is a Hopf algebra with structure maps defined by:

g=g⊗g g=1 Sg=g x_i=x_i⊗g+1⊗x_i x_i=0 Sx_i=gx_i

Radford (1990) showed that the group of Hopf automorphisms ofA_n is isomorphic to the group GL_nkof invertible n×n-matrices with coefficients ink. This group automorphism GL_nk→Aut_HopfA_nis given by:

g=g and x_i= ⁿ

k=1

_kix_k for any=_ij∈GL_nk The Hopf algebraB_n=A^cop_n is thek-algebra generated by the symbolsh y₁ y_n which satisfy the relations h²=1, y²_i =0, hy_i= −y_ih, and y_iy_j= −y_jy_i. Its Hopf algebra structure is given by:

h=h⊗h h=1 Sh=h y_i=y_i⊗1+h⊗y_i y_i=0 Sy_i= −hy_i

Let us denote the cardinality of a setTbyT. The elementsg^kx_S(resp.h^ky_S), where k∈01 and S⊆n, form a basis for A_n (resp. B_n). Since is multiplicative, it follows that

g^kx_S=

T⊆S

_TSg^kx_T⊗g^k+Tx_S\T (4.8) and

h^ky_S=

T⊆S

_TSh^k+Ty_S\T⊗h^ky_T (4.9) where_TS= ±1 and_∅S=1=_SS.

By Section 2.1, there exists a (unique) Hopf pairing A_n×B_n→k such that g h= −1, g y_j=x_i h=0, and x_i y_j=_ij for all 1≤i, j≤n.

Using (4.8) and (4.9), one gets (by induction onS) that g^kx_S h^ly_T=−1^kl_ST

for any k l∈01 andS T ⊆n, where _SS=1 and _ST =0 if S=T. Set z₀= 1+h/2 andz₁=1−h/2. The elementsz_ky_S, wherek∈01andS⊆n, form a basis forB_n such that:

g^kx_S z_ly_T=_klST (4.10) for anyk l∈01andS T ⊆n. Therefore the pairingis non-degenerate. Note that this implies thatA^∗_nA_n as a Hopf algebra.

By Theorem 2.6, we get a quasitriangular Hopf GL_nk-coalgebra DA_n B_n . For any =_ij∈GL_nk, DA_n B_n is the algebra generated by g,h, x₁ x_n,y₁ y_n, subject to the relations h²=1, (4.1), (4.2) withg replaced byh, and the following relations:

gh=hg gy_j= −y_jg hx_i= −x_ih (4.11) x_iy_j−y_jx_i=_jig−_ijh (4.12) IndeedDA_n B_n is the free algebra generated by the algebrasA_n andB_nwith cross relation (2.5). Further, it suffices to require the cross relations (2.5) for1⊗b· a⊗1witha=g x_iandb=h y_j. To simplify the notations, we identify ofawith a⊗1 and bwith 1⊗b (recall that these natural maps A_n→DA_n B_n and B_n→DA_n B_n are algebra monomorphisms). For example, leta=x_i and b=y_j. Sincex_i1=g y_j=x_i h=1 y_j=0, relation (2.5) gives

y_jx_i=x_i y_jhg1g·1+1 hg1x_i·y_j+1 hx_i y_j1·h Inserting the valuesg1=1 h=1,x_i y_j=_ij, andx_i y_jh= −_ji, we get (4.12).

From Theorem 2.3, we obtain that the comultiplication , the counit, the antipodeS, and the crossing ofDA_n B_n are given by

g=g⊗g h=h⊗h (4.13)

x_i=1⊗x_i+

n k=1

kix_k⊗g y_i=y_i⊗1+h⊗y_i (4.14)

g=h=1 x_i=y_i=0 Sg=g (4.15)

Sh=h Sx_i= ⁿ

k=1

_kigx_k Sy_i= −hy_i (4.16)

g=g h=h x_i= ⁿ

k=1

_kix_k y_i= ⁿ

k=1

˜

_iky_k (4.17) where˜_ij=⁻¹.

For any ∈GL_nk, let I be the ideal of DA_n B_n generated by g−h. Using the above description of the structure maps of DA_n B_n , we get that I=I_∈ is a crossed Hopf GL_nk-coideal of DA_n B_n . The quotient DA_n B_n /I =DA_n B_n /I_∈GLnk is precisely