V. Quantum Groups

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

Last update: 22 October 2012

The definition of the quantum group and the uniqueness theorem, Theorem (1.4), are stated in [Dri1987] §6 Example 6.2. Theorem (1.4) appears with proof in [SSt1993] Theorem 11.4.1. The statements in (3.3) and (3.4) can be found in [CPr1994] 9.2.1 and 9.3.1 and the treatment there gives references for where to find the proofs.

Definition, uniqueness, and existence

Making the Cartan matrix symmetric

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let $C={\left({\alpha }_j\left(H_i\right)\right)}_{1\le i,j\le r}$ be the corresponding Cartan matrix. There exist unique positive integers $d_1,d_2,\dots ,d_r$ such that $\text{gcd}(d_1,\dots ,d_r)=1$ and the matrix ${\left(d_i{\alpha }_j\left(H_i\right)\right)}_{1\le i,j\le r}$ is symmetric. The integers $d_1,d_2,\dots ,d_r$ are given explicitly by

$$\begin{array}{cc}\begin{array}{c}{A}_r,D_r,\\ E_6,E_7,E_8:\end{array}& d_i=1\text{for all}1\le i\le r,\\ B_r:& d_i=1\text{for}1\le i\le r-1,\text{and}{d}_r=2,\\ C_r:& d_i=2,\text{for}1\le i\le r-1,\text{and}{d}_r=1,\\ F_4:& d_1=d_2=1,\text{and}{d}_3=d_4=2,\\ G_2:& d_1=3,\text{and}{d}_2=1\text{.}\end{array}$$

The Poisson homomorphism $\delta$

Let $\delta :𝔤\to 𝔤\otimes 𝔤$ be the $ℂ\text{-linear}$ map given by

$$\delta \left(H_i\right)=0,\delta \left(X_i^{\pm }\right)=d_i(X_i^{\pm }\otimes H_i-H_i\otimes X_i^{\pm }),1\le i\le r\text{.}$$

There is a unique extension of the map $\delta :𝔤\to 𝔤\otimes 𝔤$ to a $ℂ\text{-linear}$ map $\delta :𝔘𝔤\to 𝔘𝔤\otimes 𝔘𝔤$ such that

$$\delta \left(xy\right)=\Delta \left(x\right)\delta \left(y\right)+\delta \left(y\right)\Delta \left(y\right),\text{for all}x,y\in 𝔘𝔤\text{.}$$

The definition of the quantum group

A Drinfel'd-Jimbo quantum group ${𝔘}_h𝔤$ corresponding to $𝔤$ is a deformation of $𝔘𝔤$ as a Hopf algebra over $ℂ$ such that

1. Poisson condition: $$\frac{{\Delta }_h\left(a\right)-{\Delta }_h^{\text{op}}\left(a\right)}{h}\left(\text{mod}h\right)=\delta \left(a\text{mod}h\right),\text{for all}a\in {𝔘}_h𝔤\text{.}$$ (If ${\Delta }_h\left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}$ then ${\Delta }_h^{\text{op}}\left(a\right)={\sum }_a{a}_{\left(2\right)}\otimes a_{\left(1\right)}\text{.)}$
2. Cartan subalgebra condition:

   There is a subalgebra ${𝔘}_h𝔥\subseteq {𝔘}_h𝔤$ such that

   1. ${𝔘}_h𝔥$ is a cocommutative, i.e. ${\Delta }_h\left(a\right)={\Delta }_h^{\text{op}}\left(a\right),$ for all $a\in {𝔘}_h𝔥,$
   2. The mapping ${𝔘}_h𝔥/h{𝔘}_h𝔥\to 𝔘𝔥$ is injective with image $𝔘𝔥\text{.}$

3. Cartan involution condition:

   There is a mapping $\theta :{𝔘}_h𝔤\to {𝔘}_h𝔤$ such that

   1. ${\theta }^2={\text{id}}_{{𝔘}_h𝔤},$
   2. $\theta \left({𝔘}_h𝔥\right)={𝔘}_h𝔥,$
   3. $\theta$ is an algebra homomorphism and a coalgebra antihomomorphism, i.e. $$\begin{array}{ccc}\theta \left(ab\right)& =& \theta \left(a\right)\theta \left(b\right),\text{for all}a,b\in {𝔘}_h𝔤,\text{and}\\ {\Delta }_h\left(\theta \left(a\right)\right)& =& (\theta \times \theta ){\delta }_h^{\text{op}}\left(a\right),\text{for all}a\in {𝔘}_h𝔤,\end{array}$$
   4. $\theta \text{mod}h$ is the Cartan involution.

Uniqueness of the quantum group

Let $𝔤$ be a finite dimensional simple Lie algebra. The Drinfel'd-Jimbo quantum group ${𝔘}_h𝔤$ corresponding to $𝔤$ is unique (up to equivalence of deformations).

Definition of $q\text{-integers}$ and $q\text{-factorials}$

For any symbol $q$ define

$$\begin{array}{c}{\left[n\right]}_q=\frac{q^n-q^{-n}}{q-q^{-1}},{\left[n\right]}_q!={\left[n\right]}_q{[n-1]}_q\dots {\left[2>\right]}_q{\left[1>\right]}_q,\text{and}\\ {\left[\genfrac{}{}{0ex}{}{m}{n}\right]}_q=\frac{{\left[m\right]}_q!}{{\left[n\right]}_q!{[m-n]}_q!},\text{for all positive integers}m\ge n,\end{array}$$

Presentation of the quantum group by generators and relations

Note the similarities (and the differences) between the following presentation of the quantum group by generators and relations and the presentation of the enveloping algebra of $𝔤$ given in II (2.2).

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let $C={\left({\alpha }_j\left(H_i\right)\right)}_{1\le i,j\le r}$ be the corresponding Cartan matrix. The Drinfel'd-Jimbo quantum group ${𝔘}_h𝔤$ corresponding to $𝔤$ can be presented as the algebra over $ℂ\left[\left[h\right]\right]$ generated (as a complete $ℂ\left[\left[h\right]\right]\text{-algebra}$ in the $h\text{-adic}$ topology) by

$$X_1^{-},X_2^{-},\dots ,X_r^{-},H_1,H_2,\dots ,H_r,X_1^{+},X_2^{+},\dots ,X_r^{+},$$

with relations

$$\begin{array}{cc}[H_i,H_j]=0,& \text{for all}1\le i,j\le r,\\ \begin{array}{c}[H_i,X_j^{+}]={\alpha }_j\left(H_i\right)X_j^{+},\\ [H_i,X_j^{-}]=-{\alpha }_j\left(H_i\right)X_j^{-},\end{array}& \text{for all}1\le i,j\le r,\\ [X_i^{+},X_j^{-}]={\delta }_{ij}\frac{e^{d_{i}h{H}_i}-e^{-d_{i}h{H}_i}}{e^{d_{i}h}-e^{-d_{i}h}},& \text{for}1\le i,j\le r,\\ \sum _{s+t=1-{\alpha }_j\left(H_i\right)}{(-1)}^s\left[\genfrac{}{}{0ex}{}{1-{\alpha }_j\left(H_i\right)}{s}\right]{\left(X_i^{\pm }\right)}^s{X}_j^{\pm }{\left(X_i^{\pm }\right)}^t=0,& \text{for}i\ne j,\end{array}$$

and with Hopf algebra structure given by

$$\begin{array}{c}{\Delta }_h\left(H_i\right)=H_i\otimes 1+1\otimes H_i,\\ {\Delta }_h\left(X_i^{+}\right)X_i^{+}\otimes e^{d_{i}h{H}_i}+1\otimes X_i^{+},{\Delta }_h\left(X_i^{-}\right)=X_i^{-}\otimes 1+e^{-d_{i}h{H}_i}\otimes X_i^{-},\\ S_h\left(H_i\right)=-H_i,S_h\left(X_i^{+}\right)=-X_i^{+}{e}^{-d_{i}h{H}_i},S_h\left(X_i^{-}\right)=-e^{d_{i}h{H}_i}{X}_i^{-},\\ {\epsilon }_h\left(H_i\right)={\epsilon }_h\left(X_i^{+}\right)={\epsilon }_h\left(X_i^{-}\right)=0,\end{array}$$

Cartan subalgebra $𝔘𝔥\left[\left[h\right]\right]\subseteq {𝔘}_h𝔤,$ and Cartan involution $\theta :{𝔘}_h𝔤⟶{𝔘}_h𝔤$ determined by

$$\theta \left(X_i^{+}\right)=-X_i^{-},\theta \left(X_i^{-}\right)=-X_i^{+},\theta \left(H_i\right)=-H_i\text{.}$$

The rational form of the quantum group

The rational form of the quantum group is an algebra which is similar to the algebra ${𝔘}_h𝔤$ except that it is over an arbitrary field $k\text{.}$ There are two reasons for introducing this algebra.

1. In the case when $k=ℂ\left(q\right)$ is the field this new algebra $U_q𝔤$ has "integral forms" which can be used to specialize $q$ to special values.
2. In the case when $k=ℂ$ and $q$ is a power of a prime then part of this algebra appears naturally as a Hall algebra of representations of quivers or, equivalenty, as a Grothendieck ring of $G\text{-equivariant}$ sheaves on certain varieties $E_V\text{.}$

Definition of the rational form of the quantum group

Many authors use the following form $U_q𝔤$ of the quantum group as the definition of the quantum group.

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let $C={\left({\alpha }_j\left(H_i\right)\right)}_{1\le i,j\le r}$ be the corresponding Cartan matrix. Let $k$ be a field and let $q\in k$ be an nonzero element of $k\text{.}$ The rational form of the Drinfel'd-Jimbo quantum group $U_q𝔤$ corresponding to $𝔤$ is the algebra $U_q𝔤$ over $k$ generated by

$$F_1,F_2,\dots ,F_r,K_1,K_2,\dots ,K_r,K_1^{-1},K_2^{-1},\dots K_r^{-1},E_1,E_2,\dots ,E_r,$$

with relations

$$\begin{array}{cc}{K}_i{K}_j=K_j{K}_i,& \text{for all}1\le i,j\le r,\\ K_i{K}_i^{-1}=K_i^{-1}{K}_i=1,& \text{for all}1\le i\le r,\\ \begin{array}{c}{K}_i{E}_j{K}_i^{-1}=q^{d_i{\alpha }_j\left(H_i\right)}{E}_j,\\ K_i{F}_j{K}_i^{-1}=q^{-d_i{\alpha }_j\left(H_i\right)}{F}_j,\end{array}& \text{for all}1\le i,j\le r,\\ E_i{F}_j-F_j{E}_i={\delta }_{ij}\frac{K_i-K_i^{-1}}{q^{d_i}-q^{-d_i}}& \text{for}1\le i,j\le r,\\ \sum _{s+t=1-{\alpha }_j\left(H_i\right)}{(-1)}^s\left[\genfrac{}{}{0ex}{}{1-{\alpha }_j\left(H_i\right)}{s}\right]E_i^s{E}_j{E}_i^t=0,& \text{for}i\ne j,\\ \sum _{s+t=1-{\alpha }_j\left(H_i\right)}{(-1)}^s\left[\genfrac{}{}{0ex}{}{1-{\alpha }_j\left(H_i\right)}{s}\right]F_i^s{F}_j{F}_i^t=0,& \text{for}i\ne j,\end{array}$$

and with Hopf algebra structure given by

$$\begin{array}{c}\Delta \left(K_i\right)=K_i\otimes K_i,\Delta \left(E_i\right)=E_i\otimes K_i+1\otimes E_i,\Delta \left(F_i\right)=F_i\otimes 1+K_i^{-1}\otimes F_i,\\ S\left(K_i\right)=K_i^{-1},S\left(E_i\right)=-E_i{K}_i^{-1},S\left(F_i\right)=-K_i{F}_i,\\ \epsilon \left(K_i\right)=1,\epsilon \left(E_i\right)=0,\epsilon \left(F_i\right)=0\text{.}\end{array}$$

It is very common to take $q$ to be an indeterminate and to let $k=ℂ\left(q\right)$ be the field of rational functions in $q\text{.}$

Relating the rational form and the original form of the quantum group

The relations in the rational form of the quantum group are obtained from the relations in the presentation of ${𝔘}_h𝔤$ by making the following replacements:

$$e^h⟶q,e^{h{d}_i{H}_i}⟶K_i,X_i^{-}⟶F_i,X_i^{+}⟶E_i\text{.}$$

The ring $U_q𝔤$ is an algebra over $k$ and $q\in k$ while the ring ${𝔘}_h𝔤$ is an algebra over $ℂ\left[\left[h\right]\right]$ where $h$ is an indeterminate. They have many similar properties. Most of the theorems about the structure of the algebra ${𝔘}_h𝔤$ have analogues for the case of the algebra $U_q𝔤\text{.}$ The category of modules for $U_q𝔤$ is very similar to the category of module for the enveloping algebra $𝔘𝔤\text{.}$ One should note, however, in contrast to Chapt. VI Theorem (1.1) which says that ${𝔘}_h𝔤\cong 𝔘𝔤\left[\left[h\right]\right],$ it is not true that $U_q𝔤$ is isomorphic to $𝔘𝔤,$ even if $k=ℂ$ and $q\in k\text{.}$ This fact complicates many of the proofs when one is trying to generalize results from the classical case of $𝔘𝔤$ to the quantum case $U_q𝔤\text{.}$

Integral forms of the quantum group

There are two different commonly used integral forms of a $ℂ\left(q\right)\text{-algebra}$ $U_q𝔤,$ the "non restricted integral form" $U_{𝒜}𝔤$ and the "restricted integral form" $U_{𝒜}^{\text{res}}𝔤\text{.}$ Let us begin by defining integral forms precisely.

Definition of integral forms

Let $q$ be an indeterminate and let $U_q$ be an algebra over $ℂ\left(q\right),$ the field of rational functions in $q\text{.}$ An integral form of $U_q$ is a $𝒜=\mathbb{Z}[q,q^{-1}]$ subalgebra $U_{𝒜}$ of $U_q$ such that the map

$$U_{𝒜}{\otimes }_{𝒜}ℂ\left(q\right)⟶U_q$$

is an isomorphism of $ℂ\left(q\right)$ algebras. In other words, upon extending scalars from $\mathbb{Z}[q,q^{-1}]$ to $ℂ\left(q\right)$ the algebra $U_{𝒜}$ turns into $U_q\text{.}$

Motivation for integral forms

The purpose of defining integral forms of algebras is that we can use them to specialize the variable $q$ to certain elements of $\mathbb{Q},$ or $ℝ,$ or $ℂ,$ etc. Let $U_{𝒜}$ be an integral form of an algebra $U_q$ over $ℂ\left(q\right)$ and let $\eta \in ℂ,$ $\eta \ne 0\text{.}$ The specialization at $q=\eta$ (over $ℂ\text{)}$ of $U_{𝒜}$ is the algebra over $ℂ$ given by

$$U_{\eta }=U_{𝒜}{\otimes }_{𝒜}ℂ,\text{where the equation}qc=\eta c$$

describes how $ℂ$ is an $𝒜=\mathbb{Z}[q,q^{-1}]\text{-module.}$ Similarly, we can define specializations of $U_{𝒜}$ over any field. With this last definition in mind we see that one could regard an integral form of $U_q$ as an $𝒜=\mathbb{Z}[q,q^{-1}]$ subalgebra $U_{𝒜}$ such that $U_q$ is the specialization of $U_{𝒜}$ over $ℂ\left(q\right)$ at $q=q\text{.}$

Definition of the non-restricted integral form of the quantum group

Let $q$ be an indeterminate and let $k=ℂ\left(q\right)$ be the field of rational functions in $q\text{.}$ Let $U_q𝔤$ be the corresponding rational form of the quantum group. For each $1\le i\le r,$ define elements

$${[K_i;0]}_{q^{d_i}}=\frac{K_i-K_i^{-1}}{q^{d_i}-q^{-d_i}}\text{.}$$

The non-restricted integral form of $U_q𝔤$ is the $𝒜=\mathbb{Z}[q,q^{-1}]$ subalgebra $U_{𝒜}𝔤$ of $U_q𝔤$ generated by the elements

$$F_1,F_2,\dots ,F_r,K_1^{\pm 1},K_2^{\pm 1},\dots ,K_r^{\pm 1},[K_1;0],[K_2;0],\dots ,[K_r;0],E_1,E_2,\dots ,E_r\text{.}$$

The Hopf algebra structure on $U_q𝔤$ restricts to a well defined Hopf algebra structure on $U_{𝒜}𝔤\text{.}$

Definition of the restricted integral form of the quantum group

Let $q$ be an indeterminate and let $k=ℂ\left(q\right)$ be the field of rational functions in $q\text{.}$ Let $U_q𝔤$ be the corresponding rational form of the quantum group. The restricted integral form of $U_q𝔤$ is the $𝒜=\mathbb{Z}[q,q^{-1}]$ subalgebra $U_{𝒜}^{\text{res}}𝔤$ of $U_q𝔤$ generated by the elements $K_1^{\pm 1},K_2^{\pm 1},\dots ,K_r^{\pm 1},$ and the elements

$$F_i^{\left(\ell \right)}=\frac{F_i^{\ell }}{{\left[\ell \right]}_{q^{d_i}}!},\text{and}{E}_i^{\left(\ell \right)}=\frac{E_i^{\ell }}{{\left[\ell \right]}_{q^{d_i}}!},\text{for all}1\le i\le r\text{and all}\ell \ge 1\text{.}$$

(The notation for the $q\text{-factorials}$ is as in (1.5).) The Hopf algebra structure on $U_q𝔤$ restricts to a well defined Hopf algebra structure on $U_{𝒜}^{\text{res}}𝔤\text{.}$ It is nontrivial to prove that $U_{𝒜}^{\text{res}}𝔤$ is an integral form on $U_q𝔤\text{.}$

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

page history