Drinfeld-Jimbo quantum groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 29 November 2011

Presentation of the Drinfeld-Jimbo quantum group

Let $ℂ\left[\right[h\left]\right]$ be the ring of formal power series in an indeterminate $h$. By definition $$e^x=\sum _{k\in {\mathbb{Z}}_{\ge 0}}\frac{x^k}{k!},\text{and define}q=e^{h/2}\text{and}{q}_i=q^{d_i}.$$ For each positive integer $n$ define $$\left[n\right]=\frac{q^n-q^{-n}}{q-q^{-1}},\left[n\right]!=\left[n\right][n-1]\cdots \left[2\right]\left[1\right],\left[0\right]!=1,$$ and $$\left[\genfrac{}{}{0ex}{}{n}{k}\right]=\frac{\left[n\right]!}{\left[k\right]![n-k]!}\text{for}k=1,\dots ,n.$$

Let $𝔤$ be a finite dimensional complex semisimple Lie algebra. Let $𝔥$ be the Cartan subalgebra of $𝔤$. Let ${\alpha }_i\in {𝔥}^{*}$ be the simple roots and let $H_i={\alpha }_i^{\vee }$ be the simple coroots so that the Cartan matrix is given by $$(⟨{\alpha }_i,{\alpha }_j^{\vee }⟩)=\left(a_{ij}\right)=A.$$

Let $U_h𝔤$ be the associative algebra with $1$ over $ℂ\left[\right[h\left]\right]$ generated (as an algebra complete in the $h$-adic topology) by the space $𝔥$ and the elements $X_1,\dots ,X_r$, $Y_1,\dots ,Y_r$, with relations $$\begin{array}{ll}[a_1,a_2],& \text{for}{a}_1,a_2\in 𝔥,\\ [a,X_j]={\alpha }_j\left(a\right)X_j,& \text{for}a\in 𝔥,\\ [a,Y_j]=-{\alpha }_j\left(a\right)Y_j,& \text{for}a\in 𝔥,\end{array}$$ $$\begin{array}{ll}\sum _{s+t=1-a_{ij}}{(-1)}^t\left[\genfrac{}{}{0ex}{}{1-a_{ji}}{s}\right]X_i^s{X}_j{X}_i^t=0,& \text{for}i\ne j,\\ \sum _{s+t=1-a_{ij}}{(-1)}^t\left[\genfrac{}{}{0ex}{}{1-a_{ji}}{s}\right]Y_i^s{Y}_j{Y}_i^t=0,& \text{for}i\ne j.\end{array}$$ There is a Hopf algebra structure on $U_h𝔤$ given by $$\begin{array}{ll}\Delta \left(X_i\right)=X_i\otimes e^{\left(h/4\right)H_i}+e^{-\left(h/4\right)H_i}\otimes X_i,& \\ \Delta \left(Y_i\right)=Y_i\otimes e^{\left(h/4\right)H_i}+e^{-\left(h/4\right)H_i}\otimes Y_i,& \\ \epsilon \left(X_i\right)=\epsilon \left(Y_i\right)=\epsilon \left(a\right)=0,& \text{for}a\in 𝔥,\\ S\left(X_i\right)=-e^{h/2}{X}_i,S\left(Y_i\right)=-e^{-h/2}{Y}_i,S\left(a\right)=-a,& \text{for}a\in 𝔥.\end{array}$$ Given the definition of the coproduct $\Delta$ the formulas for the counit $\epsilon$ 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 $\mathbb{Z}$ grading on the algebra $U_h𝔤$ determined by defining $$\begin{array}{ll}\mathrm{deg}\left(a\right)=0,& \text{for}a\in 𝔥.\\ \mathrm{deg}\left(X_i\right)=1,\mathrm{deg}\left(Y_i\right)=-1,& \text{for}i=1,\dots ,r.\end{array}$$ Let $U_h^{\ge 0}$ be the subalgebra of $U_h𝔤$ generated by $𝔥$ and $X_1,\dots ,X_r$. Similarly let $U_h^{\le 0}$ be the subalgebra of $U_h𝔤$ generated by $𝔥$ and $Y_1,\dots ,Y_r$. Let ${\stackrel{\sim }{H}}_1,\dots ,{\stackrel{\sim }{H}}_r$ be an orthonormal basis of $𝔥$. The algebra $U_h𝔤$ is a quasitriangular Hopf algebra and the element $ℛ$ can be written in the form, see [D, Sect. 4], $$ℛ=\exp \left(\frac{h}{2}{\gamma }_0\right)+\sum a_i^{+}\otimes b_i^{-},\text{where}{\gamma }_0=\sum _{i=1}^r{\stackrel{\sim }{H}}_i\otimes {\stackrel{\sim }{H}}_i$$ and $a_i^{+}\in U_h^{\ge 0}$, $b_i^{-}\in U_h^{\le 0}$ are homogeneous elements of degrees $\ge 1$ and $\le -1,\; respectively.$

The Cartan involution for the Drinfeld-Jimbo quantum group

In Drinfeld's ICM article [D], last part of Example 6.2, he explains that $\theta$ is an algebra automorphism, and a coalgebra antiautomorphism, ${\theta }^2=\mathrm{id}$, $\theta \left(X_i^{\pm }\right)=-X_i^{\mp }$, $\theta {|}_{𝔥}=-\mathrm{id}$, and $\theta modh$ 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 $\omega$ 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 $n\times n$ Cartan matrix and let $d_1,\dots ,d_n\in {\mathbb{Z}}_{>0}$ be minimal such that $\mathrm{diag}(d_1,\dots ,d_n)\cdot A$ is symmetric,

$$A={(⟨{\alpha }_i,{\alpha }_j^{\vee }⟩)}_{1\le i,j\le n},\text{and}{d}_i=\frac{2}{⟨{\alpha }_i,{\alpha }_i^{\vee }⟩}.$$

THIS SENTENCE IS BEAUTIFULLY AND COMPACTLY DONE IN DRINFELD's 1985 PAPER. MAYBE FOLLOW THAT?

The quantum group is the algebra $U$ generated by $$E_1,\dots ,E_n,F_1,\dots ,F_n,\text{and}{K}_1^{\pm 1},\dots ,K_n^{\pm 1},$$ with relations $$K_i^{-1}{K}_i=K_i{K}_i^{-1}=1,K_i{K}_j=K_j{K}_i,$$ $$K_i{E}_j{K}_i^{-1}=q_i^{⟨{\alpha }_i,{\alpha }_j^{\vee }⟩}{E}_j,K_i{F}_j{K}_i^{-1}=q_i^{-⟨{\alpha }_i,{\alpha }_j^{\vee }⟩}{F}_j,E_i{F}_j-F_j{E}_i={\delta }_{ij}\frac{K_i-K_i^{-1}}{q_i-q_i^{-1}},$$ $$\sum _{s=0}^{1-⟨{\alpha }_i,{\alpha }_j^{\vee }⟩}{(-1)}^s{E}_i^{(1-⟨{\alpha }_i,{\alpha }_j^{\vee }⟩-s)}{E}_j{E}_i^{\left(s\right)}=0,\text{for}i\ne j,$$ $$\sum _{s=0}^{1-⟨{\alpha }_i,{\alpha }_j^{\vee }⟩}{(-1)}^s{F}_i^{(1-⟨{\alpha }_i,{\alpha }_j^{\vee }⟩-s)}{F}_j{F}_i^{\left(s\right)}=0,\text{for}i\ne j,$$where $$E_i^{\left(r\right)}=\frac{E_i^r}{\left[r\right]!},F_i^{\left(r\right)}=\frac{F_i^r}{\left[r\right]!},\text{for}r\in {\mathbb{Z}}_{>0}.$$ Letting $$\begin{array}{rl}{\left[\genfrac{}{}{0ex}{}{K_i;c}{k}\right]}_{q_i}& =\prod _{j=1}^k\frac{K_i{q}_i^{c-(j-i)}-K_i^{-1}{q}_i^{-c-(j-i)}}{q_i^j-q_i^{-j}}\\ & =\left(\frac{K_i{q}_i^c-K_i^{-1}{q}_i^{-c}}{q_i^1-q_i^{-1}}\right)\cdots \left(\frac{K_i{q}_i^{c-(k-1)}-K_i^{-1}{q}_i^{-(c-(k-1\left)\right)}}{q_i^k-q_i^{-k}}\right)\end{array}$$ 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???), $$E_i^{\left(r\right)}{F}_i^{\left(s\right)}=\sum _{k=0}^{\min (r,s)}{F}_i^{(s-k)}{\left[\genfrac{}{}{0ex}{}{K_i;2k-s-r}{k}\right]}_{q_i}{E}_i^{(r-k)}$$

The restricted integral form of $U$ is the $\mathbb{Z}[q,q^{-1}]$-subalgebra $𝕌$ of $U$ generated by $$q^{\pm 1},E_i^{\left(r\right)},E_i^{\left(r\right)},\text{and}{K}_i^{\pm 1},$$ for $i=1,\dots ,n$ and $r\in {\mathbb{Z}}_{\ge 0}$ (IS THE INDEXING RIGHT? DO THE ${\displaystyle \left[\genfrac{}{}{0ex}{}{K_i;0}{j}\right]}$ NEED TO BE INCLUDED AS GENERATORS?)

The algebra $U$ is a Hopf algebra with $$\Delta \left(E_i\right)=E_i\otimes 1+K_i\otimes E_i,\Delta \left(F_i\right)=F_i\otimes K_i^{-1}+1\otimes F_i,\Delta \left(K_i\right)=K_i\otimes K_i,$$ $$S\left(E_i\right)=-K_i^{-1}{E}_i,S\left(F_i\right)=-F_i{K}_i,S\left(K_i\right)=K_i^{-1},$$ $$\epsilon \left(E_i\right)=0,\epsilon \left(E_i\right)=0,\epsilon \left(K_i\right)=1.$$

Since $(E_i\otimes 1)(K_i\otimes E_i)=q^{-2}(K_i\otimes E_i)(E_i\otimes 1)$, the $q$-version of the binomial theorem gives $$\Delta \left(E_i^{\left(r\right)}\right)=\sum _{k=0}^r{q}_i^{-k(r-k)}{E}_i^{\left(k\right)}\otimes K_i^k{E}_i^{(r-k)},\Delta \left(F_i^{\left(r\right)}\right)=\sum _{k=0}^r{q}_i^{k(r-k)}{F}_i^{\left(k\right)}\otimes F_i^{(r-k)}{K}_i^{-k},$$ $$S\left(E_i^{\left(r\right)}\right)={(-1)}^r{q}_i^{r(r-1)}{K}_i^{-r}{E}_i^{\left(r\right)},S\left(F_i^{\left(r\right)}\right)={(-1)}^r{q}_i^{-r(r-1)}{F}_i^{\left(r\right)}{K}_i^r,$$ $$\epsilon \left(E_i^{\left(r\right)}\right)=0,\epsilon \left(F_i^{\left(r\right)}\right)=0,\Delta \left(K_i\right)=K_i\otimes K_i,S\left(K_i\right)=K_i^{-1},\epsilon \left(K_i\right)=1,$$ and hence the subalgebra $𝕌$ is a Hopf subalgebra of $U$. BETTER CHECK THESE!

PUT THE REST OF THE RELATIONS ?? TO MAKE A PRESENTATION OF THE INTEGRAL FORM?? A REF TO CHARI AND PRESSLEY?

The subalgebra $𝕌$ also has a triangular decomposition $$𝕌={𝕌}^{-}{𝕌}_0{𝕌}^{+},$$ where

${𝕌}^{+}$ is the subalgebra of $𝕌$ generated by $E_i^{\left(r\right)}$,
${𝕌}_0$ is the subalgebra of $𝕌$ generated by $K_i^{\pm 1}$ and ${\displaystyle \left[\genfrac{}{}{0ex}{}{K_i;0}{j}\right]}$, and
${𝕌}^{-}$ is the subalgebra of $𝕌$ generated by $F_i^{\left(r\right)}$.

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 ???].

References

[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

page history