Twisted quantum doubles

PII. S016117120430236X http://ijmms.hindawi.com © Hindawi Publishing Corp.

TWISTED QUANTUM DOUBLES

DAIJIRO FUKUDA and KEN’ICHI KUGA

Received 26 February 2003

Dedicated to Professor Yukio Matsumoto’s sixtieth birthday

Using diagrammatic pictures of tensor contractions, we consider a Hopf algebra(Aop⊗᏾λ

A∗₎∗_{twisted by an element᏾}

λ∈A∗⊗Aopcorresponding to a Hopf algebra morphismλ:

A→A. We show that this Hopf algebra is quasitriangular with the universalR-matrix coming from᏾λwhenλ2_=id

A, generalizing the quantum double construction which corresponds to the caseλ=idA.

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

1. Introduction. LetAbe a Hopf algebra over a fieldk, and᏾the canonical element in

A⊗kA∗corresponding to the identity map idA:A→A. LetA∗be the dual Hopf algebra ofA, andAopthe Hopf algebra with the multiplication opposite to that ofA, that is, if µA:A⊗A→AandSA:A→Aare the multiplication and the antipode ofA, thenAop

has the multiplicationµAopand the antipodeSAop defined byµAop(x⊗y):=µA(y⊗x)

and SAop(x):=SA−1(x)for x,y∈A, while other structures are the same. Then one can define a Hopf algebra structure on the tensor productA∗_⊗

kAop by defining the

comultiplication and the antipode twisted by᏾as

∆(ξ⊗x)=᏾41∆A ∗ 13(ξ)∆

Aop

24 (x)᏾−411, S(ξ⊗x)=᏾−1

21

SA∗(ξ)⊗S_Aop(x)᏾21.

(1.1)

Other structures, that is, multiplication, unit, and counit, are the usual ones on the tensor product.

LetD(A)=(A∗_⊗A

op)∗and letR∈D(A)⊗D(A)be the image of᏾under the

map-pingAop⊗A∗ →Aop⊗1A∗⊗1_A⊗Aop →D(A)⊗D(A). Then D(A) turns out to be a

quasitriangular Hopf algebra with the universalR-matrixR.

This is the quantum double construction due to Drinfel’d [2], and it provides an ex-plicit solution to the quantum Yang-Baxter equation (QYBE). The important example is the universalR-matrix of the quantized universal enveloping algebra Uh(g)which is obtained as the homomorphic image of the universalR-matrix of the quantum double D(U≥0

Although a QYBE a priori has infinitely many solutions, certain uniqueness results can be obtained forR-matrix of the formΣei_{⊗eias in the quantum double construction} (see [3,4]). Hence, to generalize the quantum double construction, we want to consider general elements᏾λinAop⊗kA∗corresponding to mapsλ:A→A. Following the same construction as above using᏾_λin place of᏾, we arrive atDλ(A)=(Aop⊗᏾λA∗)∗which

unfortunately fails to be a quasitriangular Hopf algebra for generalλ. The purpose of this paper is to show the following theorem.

Theorem 1.1. Ifλ is a bialgebra endomorphism of A such thatλ◦λ=id_A, then Dλ(A)=(Aop⊗᏾λA∗)∗becomes a quasitriangular Hopf algebra with the universalR -matrix given by the image of᏾λunder the mappingAop⊗A∗→Aop⊗1A∗⊗1A⊗Aop→ D(A)⊗D(A).

For a Hopf algebra example, ifAis commutative and cocommutative, then the an-tipodeSA:A→Abecomes a bialgebra isomorphism satisfyingSA◦SA=idA. Another class of examples of such order-2 bialgebra automorphisms may come from the Weyl group acting as a group of reflections on the Cartan subalgebra of a simple Lie group

g. In fact, this action lifts to an action of a finite covering of the Weyl group on the universal enveloping algebraU(g)as a group of automorphisms.

The proof would look fairly complicated in the usual formalism of tensor manipu-lations because of the abundance of tensor components and indices involved. So, we want to present the proof using diagrams representing contraction of indices of vari-ous tensors. These diagrams will make the proof more visible and understandable since duality and other symmetry properties of the arguments become rather obvious. So, we begin with writing Hopf algebra structures using these diagrams.

2. Diagrams representing Hopf algebra structures. We write fixed basis elements

ofAand the dual basis elements ofA∗_{as and respectively. Hence,Ahas a basis} {ei₌ (i)

}i∈IandA∗has the dual basis {ej=

(j)}i∈I: (i)

, (j) =δ

i j.

The multiplication µA :A⊗A→A and the unit ι= ιA : k→A are expressed as

µA( (i)

⊗(j)) =(i) (j) and ι(1)=1_A=iιiei=

r

. Hence, the diagram (k) (i) (j)

represents

the coefficientµkijinµA(ei⊗ej)=kµijkek.

The comultiplication∆A_{:A→A⊗Aand the counit=}A_{:A→kare expressed as}

∆A₍(i)₎₌ ⊗

(i)

and((i))=i₌

r

. Hence, the diagram (i)

(j) (k)represents the coefficient

∆i

jkin∆A(ei)=j,k∆ijkej⊗ek.

Associativity and unitality, coassociativity and counitality may be expressed in a form where duality is obvious:

=

   =:

  

,

q

ι

=

q

ι

= ,

=

   =:

  

,

q

=

q

= .

Compatibility of algebra and coalgebra structures is

= ,

q

=

q

q

, ι

q

q

q

The antipodeSA:A→Ais written asS

q

q

S = _ι

q

q

q

whereSis an antiautomorphism of the Hopf algebraA:

q

S = S

q

q

q

q

q

q

q

S ι

=

q

ι ,

q

q

S

=

q

.

(2.4)

In general, a linear mapf:A→Ais written asf((i))=f(i

q

A linear mapfis an algebra homomorphism when

r

f =

r

f

r

r

r

ι

=

r

ι

. (2.5)

A linear mapfis a coalgebra homomorphism when

r

f ₌

r

f

r

r

r

=

r

. (2.6)

Note that related Hopf algebrasA∗_,Aop,Aop_{, and so forth can be expressed by}

us-ing the same components of the diagram forA. For example, the multiplication µA∗

of the dual Hopf algebraA∗is the transpose of the comultiplication∆A ofAwhich is specified by the coefficients (i)

(j) (k)of∆

A_(ei_{). Hence,µ} A∗ (

(j)⊗(k)) =(j) (k). Similarly,∆ A∗

(

(i))=( (i) ), andµAop( (j)

⊗(k)) =(j)(k), and so forth.

3. Proof of the theorem. Letλ:A→Abe a bialgebra endomorphism and let᏾λ∈

A⊗A∗ _{be the element corresponding toλ:᏾λ=λ}j

iei⊗ej. Diagrammatically,᏾λ= ⊗

q

Lemma3.1. The element᏾λis an invertible element ofAop⊗A∗satisfying the follow-ing relations:

id⊗∆A∗᏾λ=᏾λ13

᏾λ12,

∆Aop⊗_id᏾

λ=᏾λ13

᏾λ23,

id⊗SA∗₋₁

᏾λ=᏾−λ1,

SAop_⊗id᏾

λ=᏾−λ1.

(3.1)

Proof.

id⊗∆A∗

᏾λ=

q

= ⊗ ⊗

q

λ

q

= λ_⊗

q

q

᏾λ12.

(3.2)

This proves the first equality.

SAop_⊗id᏾

λ᏾λ=   



q

S−1

q



q

λ    = ⊗

q

S−1

q

λ λ

q

= _⊗

q

S−1

q

q

q

q

q

q

(3.3)

Here we used S−1

q

q

q

The remaining equalities may be similarly verified.

On the tensor productA∗⊗kAopwe define the multiplicationᏹ, the comultiplication ∆λ, the unitᏵ, the counitᏱ, and the antipodeSλas follows:

ᏹ(ξ⊗x)⊗(η⊗y)=(ξ⊗x)(η⊗y)=µA∗(ξ⊗x)⊗µ_Aop(η⊗y), ∆λ(ξ⊗x)=᏾λ41∆A

∗ 13(ξ)∆

Aop 24 (x)

᏾λ−411, Ᏽ(1)=A_⊗ι

A=1A∗_⊗Aop, Ᏹ(ξ⊗x)=ξ1AA(x), Sλ(ξ⊗x)=᏾λ−1

21

SA∗(ξ)⊗S_Aop(x)᏾_λ21,

forξ,η∈A∗_{, x,y∈A} op.

(3.4)

As᏾−1

λ =S

p

q

λ fromLemma 3.1, the twisted comultiplication may be written dia-grammatically as ∆λ       (i)

⊗

(j)      = ⊗ ⊗ ⊗

(i) (j)

q

λ

q

λ

q

S−1

. (3.5) Similarly, Sλ       (i)

⊗

(j)      =

q

⊗

q

S−1

(j)

q

λ

q

λ

q

. (3.6)

Proposition3.2(see [1,2,4]). The system(A∗_{⊗Aop,ᏹ,Ᏽ,∆λ,Ᏹ, Sλ)is a Hopf} alge-bra.

Proof. This is a standard fact. More generally, if an invertible element᏾∈C⊗B satisfies the conclusion ofLemma 3.1, namely,

id⊗∆B_(᏾)=᏾

13᏾12,

∆C_{⊗id(᏾)=᏾}

13᏾23,

id⊗SB−1

(᏾)=᏾−1_{, S}C_{⊗id(᏾)=᏾}−1_, (3.7)

thenB⊗Cbecomes a Hopf algebra with the structures defined as above. The verification of Hopf algebra conditions may be a little tedious but can be done without difficulty. Hence we omit it.

LetDλ(A)=((A∗⊗᏾λAop)∗,ᏹD,ᏵD,∆D,ᏱD, SD)be the dual Hopf algebra of(A∗⊗ Aop,ᏹ,Ᏽ,∆λ,Ᏹ, Sλ)and letRλ∈Dλ(A)⊗Dλ(A)be the image of᏾λ∈A⊗A∗under the mapA⊗A∗_→A⊗1∗

A⊗1A⊗A∗→Dλ(A). We now assumeλ◦λ=idA. We want to show that(Dλ(A), RA)is quasitriangular, that is, we need to check the following conditions:

id⊗∆D_R

λ=Rλ13

Rλ12,

∆D_⊗idR

λ=Rλ13

Rλ23, Rλ∆D(x)=∆Dop(x)Rλ forx∈Dλ(A),

(3.8)

where∆Dop_{is the opposite of the comultiplication}∆D_.

Noting that the multiplicationᏹDofDλ(A)is the dual of the comultiplication∆λof A∗_{⊗Aop, whose diagrammatic description is given at the end ofSection 2, we see that}

∆Dop

     

(b)

⊗

(a)

     Rλ

=

          

⊗ ⊗ ⊗

(a)

(b)

                 

 ⊗ ⊗ ⊗

q

λ

q

ι

q

     

=

(a)

λ λ

λ λ

ι

λ S−1

S−1

(b)

ε

⊗ ⊗ ⊗

= _λ λ

λ λ

ι

λ S−1

(b) (a)

ι

S−1

S−1

(b)

ε εε

⊗ ⊗ ⊗

=

λ λ

λ S−1

⊗ ⊗ ⊗

(b) (a)

=

l

(a)

λ λ

λ λ S−1

⊗ ⊗ ⊗

(b)

=

(b) (a)

λ

λ

λ λ S−1

⊗ ⊗ ⊗

=

λ

λ ι ε

(b) (a)

⊗ ⊗ ⊗

       

here we used

q

q

q

q

q

ι        

=

λ

λ λ

(b) (a)

=

λ

λ λ

(b) (a)

⊗ ⊗ ⊗

=

λ

(b) (a)

⊗ ⊗ ⊗

by the hypothesisλ2_=id.

(3.9) The other side of the desired equality is diagrammatically

Rλ∆D     

(b)

⊗

(a)

    

= λ

λ

λ λ

λ

S−1 _S−1

ε

(a)

(b)

⊗ ⊗ ⊗

ι

.

From the symmetry between this diagram and the first one and using the duality between the associativity and coassociativity, and so forth, we can modify this diagram in exactly the same way and obtain the same final diagram as above. This completes the proof ofTheorem 1.1.

References

[1] V. Chari and A. Pressley,A Guide to Quantum Groups, Cambridge University Press, Cam-bridge, 1994.

[2] V. G. Drinfel’d,Quantum groups, Proceedings of the International Congress of Mathemati-cians, Vol. 1, 2 (Berkeley, Calif, 1986), American Mathematical Society, Rhode Island, 1987, pp. 798–820.

[3] S. M. Khoroshkin and V. N. Tolstoy,The uniqueness theorem for the universalR-matrix, Lett. Math. Phys.24(1992), no. 3, 231–244.

[4] S. Majid,Physics for algebraists: noncommutative and noncocommutative Hopf algebras by a bicrossproduct construction, J. Algebra130(1990), no. 1, 17–64.

Daijiro Fukuda: Department of Mathematics and Informatics, Faculty of Science, Chiba Univer-sity, Chiba 263-8522, Japan

E-mail address:[email protected]

Ken’ichi Kuga: Department of Mathematics and Informatics, Faculty of Science, Chiba Univer-sity, Chiba 263-8522, Japan