Drinfeld-Jimbo quantum groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 29 November 2011

Presentation of the Drinfeld-Jimbo quantum group

Let [[h]] be the ring of formal power series in an indeterminate h. By definition ex = k 0 xkk! , and define q=eh/2 and qi=q di . For each positive integer n define [n]= qn -q-n q-q-1 , [n]!= [n][n-1] [2][1], [0]!=1, and [nk] = [n]! [k]![n-k]! for k=1,,n.

Let 𝔤 be a finite dimensional complex semisimple Lie algebra. Let 𝔥 be the Cartan subalgebra of 𝔤. Let αi𝔥* be the simple roots and let Hi= αi be the simple coroots so that the Cartan matrix is given by (αi, αj ) = (aij)=A.

Let Uh𝔤 be the associative algebra with 1 over [[h]] generated (as an algebra complete in the h-adic topology) by the space 𝔥 and the elements X1,, Xr, Y1,, Yr, with relations [a1,a2] , fora1, a2𝔥, [a,Xj] =αj(a) Xj, fora𝔥, [a,Yj] =-αj(a) Yj, fora𝔥, s+t=1-a ij (-1)t [ 1-aji s ] Xis Xj Xit =0, forij, s+t=1-a ij (-1)t [ 1-aji s ] Yis Yj Yit =0, forij. There is a Hopf algebra structure on Uh𝔤 given by Δ(Xi) = Xi e(h/4) Hi + e-(h/4) Hi Xi, Δ(Yi) = Yi e(h/4) Hi + e-(h/4) Hi Yi, ε(Xi) = ε(Yi) = ε(a) =0, for a𝔥, S(Xi) =-eh/2 Xi, S(Yi) =-e-h/2 Yi, S(a) =-a, for a𝔥. Given the definition of the coproduct Δ the formulas for the counit ε and the antipode S are forced by the definitions of a Hopf algebra. THERE IS SOME DISCREPANCY BETWEEN THE FORMULAS IN [LR], THE FORMULAS IN [Dr, ICM] AND THE FORMULAS IN [Dr, 1985]. SORT THIS OUT!!!

The Drinfeld-Jimbo quantum group is quasitriangular

There is a grading on the algebra Uh𝔤 determined by defining deg(a)=0, for a𝔥. deg(Xi) =1, deg(Yi) =-1, for i=1,,r . Let Uh0 be the subalgebra of Uh𝔤 generated by 𝔥 and X1, ,Xr. Similarly let Uh0 be the subalgebra of Uh𝔤 generated by 𝔥 and Y1, ,Yr. Let H1, , Hr be an orthonormal basis of 𝔥. The algebra Uh𝔤 is a quasitriangular Hopf algebra and the element can be written in the form, see [D, Sect. 4], = exp(h2 γ0) + ai+ bi-, where γ0 =i=1r Hi Hi and ai+ Uh0, bi- Uh0 are homogeneous elements of degrees 1 and -1, respectively.

The Cartan involution for the Drinfeld-Jimbo quantum group

In Drinfeld's ICM article [D], last part of Example 6.2, he explains that θ is an algebra automorphism, and a coalgebra antiautomorphism, θ2=id, θ(Xi±) =-Xi, θ|𝔥 =-id, and θmod h is the classical Cartan involution. The uniqueness statement is not in the 1985 paper, but gets significant discussion in the [Dr] and the later papers on quasiHopf algebras.

In the paper with Tingley we used ω an algebra antiinvolution and a coalgebra involution (in order to build a contravariant form). This stabilized after the 091221direct_to_calculation version (before that there were even some signs in).

Integral form of the Drinfeld-Jimbo quantum group

Let A be an n×n Cartan matrix and let d1, , dn >0 be minimal such that diag(d1, ,dn) A is symmetric,

A= ( αi, αj ) 1i,jn , and di = 2αi ,αi .

The quantum group is the algebra U generated by E1,, En, F1,, Fn, and K1±1, , Kn±1, with relations Ki-1 Ki = Ki Ki-1 =1, Ki Kj = Kj Ki, Ki Ej Ki-1 = qi αi, αj Ej , Ki Fj Ki-1 = qi- αi, αj Fj , EiFj -FjEi = δij Ki -Ki-1 qi-qi -1 , s=0 1-αi, αj (-1)s Ei (1- αi, αj -s) Ej Ei(s) =0, for ij, s=0 1-αi, αj (-1)s Fi (1- αi, αj -s) Fj Fi(s) =0, for ij, where Ei(r) = Eir [r]!, Fi(r) = Fir [r]!, for r>0 . Letting [ Ki ; c k ] qi = j=1k Ki qic -(j-i) - Ki-1 qi-c -(j-i) qij -qi-j = ( Kiqic - Ki-1 qi-c qi1 -qi-1 ) ( Kiqi c-(k-1) - Ki-1 qi- (c-(k-1) ) qik -qi-k ) the following identity is proved first for r=1, by induction on s, and then by induction on r with a fixed s (see APPENDIX???), Ei(r) Fi(s) = k=0 min(r,s) Fi(s-k ) [ Ki ; 2k-s-r k ] qi Ei(r-k )

The restricted integral form of U is the [q, q-1]-subalgebra 𝕌 of U generated by q±1, Ei(r), Ei(r), and Ki±1, for i=1,,n and r 0 (IS THE INDEXING RIGHT? DO THE [ Ki;0j ] NEED TO BE INCLUDED AS GENERATORS?)

The algebra U is a Hopf algebra with Δ(Ei) = Ei1 + KiEi , Δ(Fi) = Fi Ki-1 + 1Fi , Δ(Ki) = KiKi , S(Ei) = -Ki-1 Ei, S(Fi) = -FiKi, S(Ki) = Ki-1 , ε(Ei) =0, ε(Ei) =0, ε(Ki) =1.

Since (Ei1) (KiEi ) = q-2 (Ki Ei) (Ei1) , the q-version of the binomial theorem gives Δ( Ei(r) ) = k=0r qi-k (r-k) Ei(k) Kik Ei (r-k) , Δ( Fi(r) ) = k=0r qik (r-k) Fi(k) Fi (r-k) Ki-k , S( Ei(r) ) = (-1)r qi r(r-1) Ki-r Ei(r) , S( Fi(r) ) = (-1)r qi -r(r-1) Fi(r) Kir , ε( Ei(r) ) =0, ε( Fi(r) ) =0, Δ(Ki) =KiKi , S(Ki) =Ki-1, ε(Ki) =1, and hence the subalgebra 𝕌 is a Hopf subalgebra of U. BETTER CHECK THESE!


The subalgebra 𝕌 also has a triangular decomposition 𝕌=𝕌- 𝕌0𝕌+, where

𝕌+ is the subalgebra of 𝕌 generated by Ei(r) ,
𝕌0 is the subalgebra of 𝕌 generated by Ki±1 and [ Ki;0j ] , and
𝕌- is the subalgebra of 𝕌 generated by Fi(r) .

Notes and References

This page provides the presentation of the Drinfeld-Jimbo quantum groups by generators and relations. The first paragraphs of this exposition follow section ??? of [LR]. In section 3 we convert the Drinfeld presentation to the alternate form which is used in ?? REFER to [CP] and [Lu]. The Cartan involution is an important part of the structure as it is a key feature in the characterization of the Drinfeld-Jimbo quantum group [THEOREM ???].


[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

[Dr] V.G. Drinfelʹd, Almost cocommutative Hopf algebras, (Russian) Algebra i Analiz 1 (1989), 30–46; translation in Leningrad Math. J. 1 (1990), 321–342. MR1025154

[LR] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: The Brauer, Birman-Wenzl and type A Iwahori-Hecke algebras, Adv. Math. 125 (1997), 1-94. MR1427801.

[Kac] V. Kac, Infinite dimensional Lie algebras, Third edition, Cambridge University Press, Cambridge 1990. xxii+400pp. ISBN: 0-521-37215-1; 0-521-46693-8, MR1104219

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