VII. Properties of quantum groups

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

Last update: 2 November 2012

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the Drinfel'd-Jimbo quantum group corresponding to $𝔤$ that was defined in V (1.3). We shall often use the presentation of ${𝔘}_h𝔤$ given in V (1.6). In this chapter we shall describe some of the structure which quantum groups have. In many cases this structure is similar to the structure of the enveloping algebra $𝔘𝔤\text{.}$

The proofs of the triangular decomposition and the grading on the quantum group given in §1 can be found in [Jan1995] 4.7 and 4.21. The proof of the statements in (2.1) and (2.3), concerning the pairing $⟨,⟩,$ can be found in [Jan1995] 6.12, 6.18, 8.28, and 6.22. The statement in (2.2) follows from Chapt I, Prop. (5.5). The theorem giving the existence and uniqueness of the $ℛ\text{-matrix}$ is stated in [Dri1990] p.329 and the uniqueness is proved there. The existence follows from (7.4); see also [Lus1993] Theorem 4.1.2. The properties of the $ℛ\text{-matrix}$ stated in (3.3) are proved in [Dri1990] Prop. 3.1 and Prop. 4.2. Proofs of the statements in the secion on the Casimir element can be found in [Dri1990] Prop 2.1, Prop 3.2 and Prop. 5.1.

Theorem (5.2a) is proved in [Jan1995] 8.15-8.17 and [Lus1993] 39.2.2. Theorem (5.2b) is a non-trivial, but very natural, extension of well known results which appear, for example, in [Jan1995] Chapt. 8. The proof is a combination of the methods used in [CPr1994] 8.2B and [Jan1995] 8.4 and a calculation similar to that in the proof of [Jan1995] Lemma 8.2. The properties of the element $T_{w_0}$ given in (5.3) are proved in the following places: The formula for $\sigma \left(T_{w_0}\right)T_{w_0}$ is proved in [CPr1994] 8.2.4; The formula for $T_{w_0}^{-1}$ is proved by a method similar to [Jan1995] 8.4; The formula for ${\Delta }_h\left(T_{w_0}\right)$ is proved in [CPr1994] 8.3.11 and the remainder of the formulas are proved in [CPr1994] 8.2.3.

The construction of the Poincaré-Birkhoff-Witt basis of ${𝔘}_h𝔤$ given in section 6 appears in detail in [Jan1995] 8.18-8.30. The statement that ${𝔘}_h𝔤$ is almost a quantum double, Theorem (7.3), appears in [Dri1987] §13, and an outline of the proof can be found in [CPr1994] 8.3. The proof of Theorem (8.4) can be gleaned from a combination of [Jan1995] 6.11 and 6.18. Both of the books [Lus1993] and [Jos1995] also contain this fact.

Triangular decomposition and grading

Triangular decomposition of ${𝔘}_h𝔤$

The triangular decomposition of the quantum group ${𝔘}_h𝔤$ is analogous to the triangular decomposition of the Lie algebra $𝔤$ and the triangular decomposition of the enveloping algebra $𝔘𝔤$ given in II (2.3) and II (4.2).

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). Define

$$\begin{array}{ccc}{𝔘}_h{𝔫}^{-}& =& \text{subalgebra of}{𝔘}_h𝔤\text{generated by}{X}_1^{-},X_2^{-},\dots ,X_r^{-},\\ {𝔘}_h𝔥& =& \text{subalgebra of}{𝔘}_h𝔤\text{generated by}{H}_1,H_2,\dots ,H_r,\\ {𝔘}_h{𝔫}^{+}& =& \text{subalgebra of}{𝔘}_h𝔤\text{generated by}{X}_1^{+},X_2^{+},\dots ,X_r^{+}\text{.}\end{array}$$

The map

$$\begin{array}{ccc}{𝔘}_h{𝔫}^{-}\otimes {𝔘}_h𝔥\otimes {𝔘}_h{𝔫}^{+}& ⟶& {𝔘}_h𝔤\\ u^{-}\otimes u^0\otimes u^{+}& ⟼& u^{-}{u}^0{u}^{+}\end{array}$$

is an isomorphism of vector spaces.

The grading on ${𝔘}_h{𝔫}^{+}$ and ${𝔘}_h{𝔫}^{-}$

The gradings on the positive part ${𝔘}_h{𝔫}^{+}$ and on the negative part ${𝔘}_h{𝔫}^{-}$ of the quantum group ${𝔘}_h𝔤$ are exactly analogous to the gradings on the positive part $𝔘{𝔫}^{+}$ and the negative part $𝔘{𝔫}^{-}$ of the enveloping algebra $𝔘𝔤$ which are given in II (4.3).

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). Let ${\alpha }_1,\dots ,{\alpha }_r$ be the simple roots for $𝔤$ and let

$$Q^{+}=\sum _i\mathbb{N}{\alpha }_i,\text{where}\mathbb{N}={\mathbb{Z}}_{\ge 0}\text{.}$$

For each element $\nu ={\sum }_{i=1}^r{\nu }_i{\alpha }_i\in Q^{+}$ define

$$\begin{array}{ccc}{\left({𝔘}_h{𝔫}^{+}\right)}_{\nu }& =& \text{span-}\{X_{i_1}^{+}\dots X_{i_p}^{+}\mid X_{i_1}^{+}\dots X_{i_p}^{+}\text{has}{\nu }_j\text{-factors of type}{X}_j^{+}\}\\ {\left({𝔘}_h{𝔫}^{-}\right)}_{\nu }& =& \text{span-}\{X_{i_1}^{-}\dots X_{i_p}^{-}\mid X_{i_1}^{-}\dots X_{i_p}^{-}\text{has}{\nu }_j\text{-factors of type}{X}_j^{-}\}\text{.}\end{array}$$

Then

$${𝔘}_h{𝔫}^{-}=\underset{\nu \in Q^{+}}{⨁}{\left({𝔘}_h{𝔫}^{-}\right)}_{\nu }\text{and}{𝔘}_h{𝔫}^{+}=\underset{\nu \in Q^{+}}{⨁}{\left({𝔘}_h{𝔫}^{+}\right)}_{\nu },$$

as vector spaces.

The inner product $⟨,⟩$

In some sense the nonnegative part ${𝔘}_h{𝔟}^{+}$ of the quantum group is the dual of the nonpositive part ${𝔘}_h{𝔟}^{-}$ of the quantum group. This reflected in the fact that there is a nondegenerate bilinear pairing on all of ${𝔘}_h𝔤\text{.}$ The extended pairing is an analogue of the Killing form on $𝔤$ in two ways:

1. it is an ad-invariant form on ${𝔘}_h𝔤,$ and
2. upon restriction to $𝔤$ it coincides $\left(\text{mod}h\right)$ with the Killing form.

The pairing between ${𝔘}_h{𝔟}^{-}$ and ${𝔘}_h{𝔟}^{+}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). Define

$$\begin{array}{ccc}{𝔘}_h{𝔟}^{-}& =& \text{subalgebra of}{𝔘}_h𝔤\text{generated by}{X}_1^{-},X_2^{-},\dots ,X_r^{-}\text{and}{H}_1,\dots ,H_r,\\ {𝔘}_h{𝔟}^{+}& =& \text{subalgebra of}{𝔘}_h𝔤\text{generated by}{X}_1^{+},X_2^{+},\dots ,X_r^{+}\text{and}{H}_1,\dots ,H_r\text{.}\end{array}$$

1. There is a unique $ℂ\left[\left[h\right]\right]\text{-bilinear}$ pairing $$⟨,⟩:{𝔘}_h{𝔟}^{-}\times {𝔘}_h{𝔟}^{+}⟶ℂ\left[\left[h\right]\right]\text{which satisfies}$$
   1. $⟨1,1⟩=1,$
   2. $⟨H_i,H_j⟩=\frac{{\alpha }_j\left(H_i\right)}{d_j},$
   3. $⟨X_i^{-},X_j^{+}⟩={\delta }_{ij}\frac{1}{e^{d_{i}h}-e^{-d_{i}h}},$
   4. $⟨ab,c⟩=⟨a\otimes b,{\Delta }_h\left(c\right)⟩,\text{for all}a,b\in {𝔘}_h{𝔟}^{-}\text{and}c\in {𝔘}_h{𝔟}^{+},$
   5. $⟨a,bc⟩=⟨{\Delta }_h^{\text{op}}\left(a\right),b\otimes c⟩,\text{for all}a\in {𝔘}_h{𝔟}^{-}\text{and}b,c\in {𝔘}_h{𝔟}^{+}\text{.}$
2. The pairing $⟨,⟩$ is nondegenerate.
3. The pairing $⟨,⟩$ respects the gradings on ${𝔘}_h{𝔫}^{+}$ and ${𝔘}_h{𝔫}^{-}$ in the following sense:
   1. Let $\mu ,\nu \in Q^{+}\text{.}$ $$\text{If}\mu \ne \nu \text{then}⟨{\left({𝔘}_h{𝔟}^{-}\right)}_{\mu },{\left({𝔘}_h{𝔟}^{+}\right)}_{\nu }⟩=0\text{.}$$
   2. Let $\nu \in Q^{+}\text{.}$ The restriction of the pairing $⟨,⟩$ to ${\left({𝔘}_h{𝔫}^{-}\right)}_{\nu }\times {\left({𝔘}_h{𝔫}^{+}\right)}_{\nu }$ is a nondegenerate pairing $$⟨,⟩:{\left({𝔘}_h{𝔫}^{-}\right)}_{\nu }\times {\left({𝔘}_h{𝔫}^{+}\right)}_{\nu }\to ℂ\left[\left[h\right]\right]\text{.}$$

If $\theta$ is the Cartan involution of ${𝔘}_h𝔤$ as given in V (1.6) and $S_h$ is the antipode of ${𝔘}_h𝔤$ then

$$⟨\theta \left(u^{+}\right),\theta \left(u^{-}\right)⟩=⟨u^{-},u^{+}⟩\text{and}⟨S_h\left(u^{-}\right),S_h\left(u^{+}\right)⟩=⟨u^{-},u^{+}⟩,$$

for all $u^{-}\in {𝔘}_h{𝔟}^{-}$ and $u^{+}\in {𝔘}_h{𝔟}^{+}\text{.}$

Extending the pairing to an ad-invariant pairing on ${𝔘}_h𝔤$

The triangular decomposition (1.1) of ${𝔘}_h𝔤$ says that ${𝔘}_h𝔤\cong {𝔘}_h{𝔫}^{-}\otimes {𝔘}_h𝔥\otimes {𝔘}_h{𝔫}^{+}$ and that

$$\text{every element}u\in {𝔘}_h𝔤\text{can be written in the form}{u}^{-}{u}^0{u}^{+},$$

where $u^{-}\in {𝔘}_h{𝔫}^{-},$ $u^0\in {𝔘}_h𝔥,$ and $u^{+}\in {𝔘}_h{𝔫}^{+}\text{.}$ We can use this to extend the pairing defined in (2.1) to a pairing

$$\begin{array}{c}⟨,⟩:{𝔘}_h𝔤\times {𝔘}_h𝔤⟶ℂ\left[\left[h\right]\right]\text{defined by the formula}\\ ⟨u_1^{-}{u}_1^0{u}_1^{+},u_2^{-}{u}_2^0{u}_2^{+}⟩=⟨u_1^{-},S_h\left(u_2^0{u}_2^{+}\right)⟩⟨u_2^{-},S_h^{-1}\left(u_1^0{u}_1^{+}\right)⟩,\end{array}$$

for all $u_1^{-},u_2^{-}\in {𝔘}_h{𝔫}^{-},$ $u_1^0,u_2^0\in {𝔘}_h𝔥,$ and $u_1^{+},u_2^{+}\in {𝔘}_h{𝔫}^{+},$ where $S_h$ is the antipode of ${𝔘}_h𝔤\text{.}$ Then

$$⟨{\text{ad}}_u\left(v_1\right),v_2⟩=⟨v_1,{\text{ad}}_{S_h\left(u\right)}\left(v_2\right)⟩,\text{for all}u,v_1,v_2\in {𝔘}_h𝔤,$$

This formula says that the extended pairing $⟨,⟩$ is an ad-invariant pairing as defined in I (5.5). The pairing $⟨,⟩:{𝔘}_h𝔤\to {𝔘}_h𝔤ℂ\left[\left[h\right]\right]$ is not symmetric, see I (5.5).

Duality between matrix coefficients for representations and $U_q𝔤\text{.}$

Let $𝔤$ be finite dimensional complex simple Lie algebra and let $U_q𝔤$ be the rational form of the quantum group over a field $k,$ where char $k\ne 2,3$ and $q\in k$ is not a root of unity. Let $Q$ be the root lattice for $𝔤\text{.}$

Let $M$ be a finite dimensional $U_q𝔤$ module such that all weights $\lambda$ of $M$ satisfy $2\lambda \in Q\text{.}$ Then, for each pair $n^{*}\in M^{*}$ and $m\in M$ there is a unique element $u\in U_q𝔤$ such that

$$n^{*}\left(vm\right)=⟨v,u⟩,\text{for all}v\in U_q𝔤,$$

where $⟨,⟩$ is the bilinear form on $U_q𝔤$ given by (2.1) after making the substitutions in V (2.2).

The function

$$c_{m,n^{*}}:U_q𝔤\to ℂ\left(q\right)\text{defined by}{c}_{m,n^{*}}\left(v\right)=⟨n^{*},vm⟩$$

is the $(m,n^{*})\text{-matrix}$ coefficient of $v$ acting on $M\text{.}$ The above theorem gives a duality between matrix coefficient functions and $U_q𝔤\text{.}$ IT also says that every element of $U_q𝔤$ is determined by how it acts on finite dimensional $U_q𝔤\text{-modules.}$

The universal $ℛ\text{-matrix}$

Motivation for the $ℛ\text{-matrix}$

The following theorem states that there is an element $ℛ$ such that the pair $({𝔘}_h𝔤,ℛ)$ is a quasitriangular Hopf algebra. In particular, this implies that the category of finite dimensional modules for the quantum group $𝔘𝔤$ is a braided SRMCwMFF.

Existence and uniqueness of $ℛ$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and ${𝔘}_h𝔤$ be the corresponding quantum group as presented in V (1.6). Recall the Killing form on $𝔤$ from II (1.6).

Let $\left\{H_i\right\}$ be an orthonormal basis of $𝔥$ with respect to the Killing form and define

$$t_0=\sum _{i=1}^r{\stackrel{\sim }{H}}_i\otimes {\stackrel{\sim }{H}}_i\text{.}$$

If $\nu \in Q^{+}$ (see (1.2)) and $\nu =\sum _{i=1}^r{\nu }_i{\alpha }_i$ where ${\alpha }_1,\dots ,{\alpha }_r$ are the simple roots, define $n_{\nu }$ to be the smallest number of positive roots $\alpha >0$ whose sum is equal to $\nu \text{.}$

The element $ℛ$ is not quite an element of ${𝔘}_h𝔤\otimes {𝔘}_h𝔤$ so we have to make the tensor product just a tiny bit bigger. To do this we let ${𝔘}_h𝔤⨶{𝔘}_h𝔤$ denote the $h\text{-adic}$ completion of the tensor product ${𝔘}_h𝔤\otimes {𝔘}_h𝔤,$ see III §1.

There exists a unique invertible element $ℛ\in {𝔘}_h𝔤⨶{𝔘}_h𝔤$ such that

$$\begin{array}{c}ℛ{\Delta }_h\left(a\right){ℛ}^{-1}={\Delta }_h^{\text{op}}\left(a\right),\text{for all}a\in {𝔘}_h𝔤,\text{and}\\ ℛ\text{has the form}ℛ=\sum _{\nu \in Q^{+}}\text{exp}\left\{h(t_0+\frac{1}{2}(H_{\nu }\otimes 1-1\otimes H_{\nu }))\right\}{P}_{\nu },\text{where}\end{array}$$ $$\begin{array}{c}{P}_{\nu }\in {\left({𝔘}_h{𝔫}^{-}\right)}_{\nu }\otimes {\left({𝔘}_h{𝔫}^{+}\right)}_{\nu },\\ H_{\nu }={\sum }_{i=1}^r{\nu }_i{H}_i,\text{if}\nu ={\sum }_i{\nu }_i{\alpha }_i,\\ P_{\nu }\text{is a polynomial in}{X}_i^{+}\otimes 1\text{and}1\otimes X_i^{-},1\le i\le r,\text{with coefficients in}ℂ\left[\left[h\right]\right],\\ \text{such that}\\ \text{the smallest power of h in}{P}_{\nu }\text{with nonzero coefficients is}{h}^{n_{\nu }}\text{.}\end{array}$$

Properties of the $ℛ\text{-matrix}$

Recall V (1.6) that ${𝔘}_h𝔤$ is a Hopf algebra with comultiplication ${\Delta }_h,$ counit ${\epsilon }_h,$ and antipode $S_h$ and that ${𝔘}_h𝔤$ comes with a Cartan involution $\theta \text{.}$ The following formulas describe the relationship between the $ℛ\text{-matrix}$ and the Hopf algebra structure on ${𝔘}_h𝔤\text{.}$ If $ℛ=\sum a_i\otimes b_i$ then let

$$\begin{array}{c}{ℛ}_{12}=\sum a_i\otimes b_i\otimes 1,{ℛ}_{13}=\sum a_i\otimes 1\otimes b_i,\text{and}{ℛ}_{23}=\sum 1\otimes a_i\otimes b_i,\\ \text{and let}{ℛ}_{21}=\sum b_i\otimes a_i\text{.}\end{array}$$

Let $\sigma :{𝔘}_h𝔤\to {𝔘}_h𝔤$ be the $ℂ\text{-algebra}$ automorphism of ${𝔘}_h𝔤$ given by $\sigma \left(h\right)=-h,$ $\sigma \left(X_i^{\pm }\right)=X_i^{\pm },$ and $\sigma \left(H_i\right)=H_i\text{.}$ With these notations we have

$$\begin{array}{c}({\Delta }_h\otimes \text{id})\left(ℛ\right)={ℛ}_{13}{ℛ}_{23},\text{and}(\text{id}\otimes {\Delta }_h)\left(ℛ\right)={ℛ}_{13}{ℛ}_{12},\\ ({\epsilon }_h\otimes \text{id})\left(ℛ\right)=1=(\text{id}\otimes {\epsilon }_h)\left(ℛ\right),\\ (S_h\otimes \text{id})\left(ℛ\right)=(\text{id}\otimes S_h^{-1})\left(ℛ\right)={ℛ}^{-1}\text{and}(S_h\otimes S_h)\left(ℛ\right)=ℛ,\\ (\theta \otimes \theta )\left(ℛ\right)={ℛ}_{21}\text{and}(\sigma \otimes \sigma )\left(ℛ\right)={ℛ}^{-1}\text{.}\end{array}$$

An analogue of the Casimir element

Definition of the element $u$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). The antipode $S_h:{𝔘}_h𝔤\to {𝔘}_h𝔤$ is an antiautomorphism of ${𝔘}_h𝔤,$ see I (2.1). This means that the map $S_h^2:{𝔘}_h𝔤\to {𝔘}_h𝔤$ is an automorphism of ${𝔘}_{𝔤}\text{.}$ The following theorem says that this automorphism is inner!

Let $ℛ\in {𝔘}_h𝔤⨶{𝔘}_h𝔤$ be the universal $ℛ\text{-matrix}$ of ${𝔘}_h𝔤$ as defined in (3.2). Suppose that $ℛ=\sum a_i\otimes b_i$ and define $u=\sum S\left(b_i\right)a_i\text{.}$ Then $u$ is invertible and

$$ua{u}^{-1}=S_h^2\left(a\right),\text{for all}a\in {𝔘}_h𝔤\text{.}$$

Properties of the element $u\text{.}$

The relationship of the element $u$ to the Hopf algebra structure of ${𝔘}_h𝔤$ is given by the formulas

$${\Delta }_h\left(u\right)={\left({ℛ}_{21}{ℛ}_{12}\right)}^{-1}(u\otimes u),S_h\left(u\right)=u,\text{and}{\epsilon }_h\left(u\right)=1,$$

where ${ℛ}_{12}=ℛ=\sum a_i\otimes b_i$ is the universal $ℛ\text{-matrix}$ of ${𝔘}_h𝔤$ given in (3.2), and ${ℛ}_{21}=\sum b_i\otimes a_i\text{.}$ The inverse of the element $u$ is given by

$$u^{-1}=\sum S_h^{-1}\left(d_j\right)c_j,\text{where}{ℛ}^{-1}=\sum c_j\otimes d_j\text{.}$$

Why the element $u$ is an analogue of the Casimir element

Let $\stackrel{\sim }{\rho }$ be the element of $𝔥$ such that ${\alpha }_i\left(\stackrel{\sim }{\rho }\right)=1$ for all simple roots ${\alpha }_i$ of $𝔤\text{.}$ An easy check on the generators of ${𝔘}_h𝔤$ shows that

$$e^{h\stackrel{\sim }{\rho }}a{e}^{-h\stackrel{\sim }{\rho }}=S_h^2\left(a\right),\text{for all}a\in {𝔘}_h𝔤\text{.}$$

It follows that

$$\text{the element}{e}^{-h\stackrel{\sim }{\rho }}u=u{e}^{-h\stackrel{\sim }{\rho }}\text{is a central element in}{𝔘}_h𝔤\text{.}$$

Any central element of ${𝔘}_h𝔤$ must act on each finite dimensional simple ${𝔘}_h𝔤\text{-module}$ by a constant. For each dominant integral weight $\lambda$ let $L\left(\lambda \right)$ be the finite dimensional simple ${𝔘}_h𝔤\text{-module}$ indexed by $\lambda$ (see VI (1.3)). As in II (4.5), let $\rho$ be the element of ${𝔥}_{ℝ}^{*}$ given by

$$\rho =\frac{1}{2}\sum _{\alpha >0}\alpha ,$$

where the sum is over all positive roots for $𝔤\text{.}$ Then the element

$$e^{-h\stackrel{\sim }{\rho }}u\text{acts on}L\left(\lambda \right)\text{by the constant}{q}^{-(\lambda +\rho ,\lambda +\rho )+(\rho ,\rho )},$$

where $q=e^h$ and the inner product in the exponent of $q$ is the inner product on ${𝔥}_{ℝ}^{*}$ given in II (2.7). Note that analogy with II (4.5). It is also interesting to note that

$${\left(e^{-h\stackrel{\sim }{\rho }}u\right)}^2=u{S}_h\left(u\right)\text{.}$$

The element $T_{w_0}$

The automorphism $\varphi \circ \theta \circ S_h$

Let $W$ be the Weyl group corresponding to $𝔤$ and let $w_0$ be the longest element of $W$ (see II (2.8)). Let $s_1,\dots ,s_r$ be the simple reflections in $W\text{.}$ For each $1\le i\le r$ there is a unique $1\le j\le r$ such that $w_0{s}_i{w}_0^{-1}=s_j\text{.}$ The map given by

$$\varphi \left(X_i^{\pm }\right)=X_j^{\pm },\text{and}\varphi \left(H_i\right)=H_j,\text{where}{w}_0{s}_i{w}_0^{-1}=s_j,\text{for}1\le i\le r,$$

extends to an automorphism of ${𝔘}_h𝔤\text{.}$ Let $\stackrel{\sim }{\theta }$ be the anti-automorphism of ${𝔘}_h𝔤$ defined by $\stackrel{\sim }{\theta }\left(X_i^{\pm }\right)=X_i^{\mp }$ and $\stackrel{\sim }{\theta }\left(H_i\right)=H_i\text{.}$ This is an analogue of the Cartan involution. Let $S_h$ be the antipode of ${𝔘}_h𝔤$ as given in V (1.6). There are both anti-automorphisms of ${𝔘}_h𝔤\text{.}$ The composition.

$$(S_h\circ \stackrel{\sim }{\theta }\circ \varphi ):{𝔘}_h𝔤\to {𝔘}_h𝔤$$

is an automorphism of ${𝔘}_h𝔤\text{.}$ The following result says that this automorphism is inner.

Definition of the element $T_{w_0}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the corresponding quantum group as presented in V (1.6). Let $q=e^h$ and for each $1\le i\le r$ let

$$E_i^{\left(r\right)}=\frac{{\left(X_i^{+}\right)}^r}{{\left[r\right]}_{q^{d_i}}!},F_i^{\left(r\right)}=\frac{{\left(X_i^{+}\right)}^r}{{\left[r\right]}_{q^{d_i}}!}\text{and}{K}_i=e^{h{d}_i{H}_i},$$

where the notation for $q\text{-factorials}$ is as in V (1.5). For each $1\le i\le r,$ define

$$T_i=\sum _{a,b,c\ge 0}{(-1)}^b{q}^{b-ac+(c+a-b)(c-a)}{E}_i^{\left(a\right)}{F}_i^{\left(b\right)}{E}_i^{\left(c\right)}{K}_i^{c-a},$$

where the sum is over all nonnegative integers $a,b,$ and $c\text{.}$

1. The elements $T_i$ satisfy the relations $$\underset{\underset{m_{ij}\text{factors}}{⏟}}{T_i{T}_j{T}_i{T}_j\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{T_j{T}_i{T}_j{T}_i\dots }\text{for}i\ne j,$$ where the $m_{ij}$ are as given in II (2.8).
2. Let $w_0=s_{i_1}\dots s_{i_N}$ be a reduced word for the longest element of the Weyl group $W,$ see II (2.8). Define $$T_{w_0}=T_{i_1}\dots T_{i_N}\text{.}$$ Then $T_{w_0}$ is invertible and $$T_{w_0}a{T}_{w_0}^{-1}=(S_h\otimes \stackrel{\sim }{\theta }\otimes \varphi )\left(a\right),\text{for all}a\in {𝔘}_h𝔤\text{.}$$

Properties of the element $T_{w_0}$

Let $u\in {𝔘}_h𝔤$ be the analogue of the Casimir element for ${𝔘}_h𝔤$ as given in §4 and let $\sigma$ be the $ℂ\text{-algebra}$ automorphism of ${𝔘}_h𝔤$ given in (3.3). Let $\stackrel{\sim }{\sigma }$ be the $ℂ\text{-linear}$ automorphism of ${𝔘}_h𝔤$ given in $\stackrel{\sim }{\sigma }\left(h\right)=-h,$ $\stackrel{\sim }{\sigma }\left(X_i^{\pm }\right)=X_i^{\mp },$ and $\stackrel{\sim }{\sigma }\left(H_i\right)=-H_i\text{.}$ Then

$$\sigma \left(T_{w_0}\right)T_{w_0}=u\text{and}{T}_{w_0}^{-1}=\stackrel{\sim }{\sigma }\left(T_{w_0}\right)\text{.}$$

The relationship between the element $T_{w_0}$ and the Hopf algebra structure of ${𝔘}_h𝔤$ is given by the formulas

$${\Delta }_h\left(T_{w_0}\right)={ℛ}_{12}^{-1}(T_{w_0}\otimes T_{w_0})=(T_{w_0}\otimes T_{w_0}){ℛ}_{21}^{-1},S_h\left(T_{w_0}\right)=T_{w_0}{e}^{h\stackrel{\sim }{\rho }},\text{and}{\epsilon }_h\left(T_{w_0}\right)=1,$$

where ${ℛ}_{12}=ℛ=\sum a_i\otimes b_i$ is the universal $ℛ\text{-matrix}$ of ${𝔘}_h𝔤$ given in (3.2), and ${ℛ}_{21}=\sum b_i\otimes a_i\text{.}$

The Poincaré-Birkhoff-Witt bais of ${𝔘}_h𝔤$

Root vectors in ${𝔘}_h𝔤$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the corresponding quantum group as presented in V (1.6). Let $T_i$ be the elements of ${𝔘}_h𝔤$ given in (5.2). Define an automorphism ${\tau }_i:{𝔘}_h𝔤\to {𝔘}_h𝔤$ by

$${\tau }_i\left(u\right)=T_{i}u{T}_i^{-1},\text{for all}u\in {𝔘}_h𝔤,$$

Let $W$ be the Weyl group corresponding to $𝔤\text{.}$ Fix a reduced decomposition $w_0=s_{i_1}\dots s_{i_N}$ of the longest element $w_0\in W,$ see II (2.8). Define

$${\beta }_1={\alpha }_{i_1},{\beta }_2=s_{i_1}\left({\alpha }_{i_2}\right),\dots ,{\beta }_N=s_{i_1}{s}_{i_2}\dots s_{i_{N-1}}\left({\alpha }_{i_N}\right)\text{.}$$

The elements ${\beta }_1,\dots ,{\beta }_N$ are the positive roots $𝔤\text{.}$ Define elements of ${𝔘}_h𝔤$ by

$$X_{{\beta }_1}^{\pm }=X_{i_1}^{\pm },X_{{\beta }_2}^{\pm }={\tau }_{i_1}\left(X_{i_2}^{\pm }\right),\dots ,X_{{\beta }_N}^{\pm }={\tau }_{i_1}{\tau }_{i_2}\dots {\tau }_{i_{N-1}}\left(X_{i_N}^{\pm }\right)\text{.}$$

These elements depend on the choice of the reduced decomposition. They are analogues of the elements $X_{\beta }$ and $X_{-\beta }$ in $𝔘𝔤$ which are given in II (4.4).

Poincaré-Birkhoff-Witt bases of ${𝔘}_h{𝔫}^{-},$ ${𝔘}_h𝔥,$ and ${𝔘}_h{𝔫}^{+}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the corresponding quantum group as presented in V (1.6). Let ${𝔘}_h{𝔫}^{-},$ ${𝔘}_h𝔥,$ and ${𝔘}_h{𝔫}^{+}$ be the subalgebras of ${𝔘}_h𝔤$ defined in (1.1). The following bases of ${𝔘}_h{𝔫}^{-},$ ${𝔘}_h𝔥,$ ${𝔘}_h{𝔫}^{+},$ and ${𝔘}_h𝔤$ are analogues of the Poincaré-Birkhoff-Witt bases of ${𝔘}_h{𝔫}^{-},$ ${𝔘}_h𝔥,$ and ${𝔘}_h{𝔫}^{+}$ which are given in II (4.4).

Let $X_{{\beta }_1}^{\pm },\dots ,X_{{\beta }_N}^{\pm }$ be the elements of ${𝔘}_h𝔤$ defined in (6.1). Then

$$\begin{array}{c}\{{\left(X_{{\beta }_1}^{+}\right)}^{p_1}{\left(X_{{\beta }_2}^{+}\right)}^{p_2}\dots {\left(X_{{\beta }_N}^{+}\right)}^{p_N}\mid p_1,\dots ,p_N\in {\mathbb{Z}}_{\ge 0}\}\text{is a basis of}{𝔘}_h{𝔫}^{+},\\ \{{\left(X_{{\beta }_1}^{-}\right)}^{n_1}{\left(X_{{\beta }_2}^{-}\right)}^{n_2}\dots {\left(X_{{\beta }_N}^{-}\right)}^{n_N}\mid n_1,\dots ,n_N\in {\mathbb{Z}}_{\ge 0}\}\text{is a basis of}{𝔘}_h{𝔫}^{-},\\ \{H_1^{s_1}{H}_2^{s_2}\dots H_1^{s_r}\mid s_1,\dots ,s_N\in {\mathbb{Z}}_{\ge 0}\}\text{is a basis of}{𝔘}_h𝔥\text{.}\end{array}$$

The PBW-bases of ${𝔘}_h{𝔫}^{-}$ and ${𝔘}_h{𝔫}^{+}$ are dual bases with respect to $⟨,⟩$ (almost)

Recall the pairing between ${𝔘}_h{𝔟}^{-}$ and ${𝔘}_h{𝔟}^{+}$ given in (2.1).

Let $w_0=s_{i_1}\dots s_{i_N}$ be a reduced decomposition of the longest element of the Weyl group and let ${\beta }_j$ and $X_{{\beta }_j}^{\pm },$ $1\le j\le N,$ be the elements defined in (6.1). Let $p_1,\dots ,p_N,$ $n_1,\dots ,n_N\in {\mathbb{Z}}_{\ge 0}\text{.}$ Then

$$⟨{\left(X_{{\beta }_1}^{-}\right)}^{n_1}{\left(X_{{\beta }_2}^{-}\right)}^{n_2}\dots {\left(X_{{\beta }_N}^{-}\right)}^{n_N},{\left(X_{{\beta }_1}^{+}\right)}^{p_1}{\left(X_{{\beta }_2}^{+}\right)}^{p_2}\dots {\left(X_{{\beta }_N}^{+}\right)}^{p_N}⟩=\prod _{j=1}^N{\delta }_{n_j,p_j}⟨{\left(X_{i_j}^{-}\right)}^{n_j},{\left(X_{i_j}^{+}\right)}^{n_{\mathrm{js}}}⟩,$$

where ${\delta }_{n_j,p_j}$ is the Kronecker delta.

Furthermore, we have that, for each $1\le i\le r,$

$$⟨{\left(X_i^{-}\right)}^n,{\left(X_i^{+}\right)}^n⟩={(-1)}^n{q}^{-d_{i}n(n-1)/2}\frac{{\left[n\right]}_{q^{d_i}}!}{{(q^{d_i}-q^{-d_i})}^n},\text{where}q=e^h\text{.}$$

The quantum group is a quantum double (almost)

The identification of ${\left({𝔘}_h{𝔟}^{+}\right)}^{*\text{coop}}$ with ${𝔘}_h{𝔟}^{-}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the corresponding quantum group as presented in V (1.6). Define

$$\begin{array}{ccc}{𝔘}_h{𝔟}^{-}& =& \text{subalgebra of}{𝔘}_h𝔤\text{generated by}{X}_1^{-},X_2^{-},\dots ,X_r^{-}\text{and}{H}_1,\dots ,H_r,\\ {𝔘}_h{𝔟}^{+}& =& \text{subalgebra of}{𝔘}_h𝔤\text{generated by}{X}_1^{+},X_2^{+},\dots ,X_r^{+}\text{and}{H}_1,\dots ,H_r,\end{array}$$

except let us distinguish the elements $H_i$ which are in ${𝔘}_h{𝔟}^{+}$ from the elements $H_i$ which are in ${𝔘}_h{𝔟}^{-}$ by writing $H_i^{+}$ and $H_i^{-}$ respectively, instead of just $H_i$ in both cases.

The nondegeneracy of the pairing $⟨,⟩$ between ${𝔘}_h{𝔟}^{+}$ and ${𝔘}_h{𝔟}^{-}$ (see (2.1)) shows that ${𝔘}_h{𝔟}^{-}$ is essentially the dual of $U_h{𝔟}^{+}\text{.}$ Furthermore, it follows from the conditions

$$⟨x_1{x}_2,y⟩=⟨x_1\otimes x_2,{\Delta }_h\left(y\right)⟩\text{and}⟨x,y_1{y}_2⟩=⟨{\Delta }^{\text{op}}\left(x\right),y_1\otimes y_2⟩$$

that the multiplication in ${𝔘}_h{𝔟}^{-}$ is the adjoint of the comultiplication in ${𝔘}_h{𝔟}^{+}$ and the opposite of the comultiplication in ${𝔘}_h{𝔟}^{-}$ is the adjoint of the multiplication in ${𝔘}_h{𝔟}^{+}\text{.}$ Thus (here we are fudging a bit since ${𝔘}_h{𝔟}^{+}$ is infinite dimensional),

$${𝔘}_h{𝔟}^{-}\simeq {\left({𝔘}_h{𝔟}^{+}\right)}^{*\text{coop}}\text{as Hopf algebras,}$$

where ${\left({𝔘}_h{𝔟}^{+}\right)}^{*\text{coop}}$ is the Hopf algebra defined in I (5.2).

Recalling the quantum double

Recall, from I (5.3), that the quantum double $D\left(A\right)$ of a finite dimensional Hopf algebra $A$ is the new Hopf algebra

$$D\left(A\right)=\{a\alpha \mid a\in A,\alpha \in A^{*\text{coop}}\}\cong A\otimes A^{*\text{coop}}$$

with multiplication determined by the formulas

$$\begin{array}{ccc}\alpha a& =& \sum _{\alpha ,a}⟨{\alpha }_{\left(1\right)},S^{-1}\left(a_{\left(1\right)}\right)⟩⟨{\alpha }_{\left(3\right)},a_{\left(3\right)}⟩a_{\left(2\right)}{\alpha }_{\left(2\right)},\text{and}\\ a\alpha & =& \sum _{\alpha ,a}⟨{\alpha }_{\left(1\right)},a_{\left(1\right)}⟩⟨{\alpha }_{\left(3\right)},S^{-1}\left(a_{\left(3\right)}\right)⟩{\alpha }_{\left(2\right)}{a}_{\left(2\right)},\end{array}$$

where, if $\Delta$ is the comultiplication in $A$ and $A^{*\text{coop}},$

$$(\Delta \otimes \text{id})\circ \Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}\otimes a_{\left(3\right)},\text{and}(\Delta \otimes \text{id})\circ \Delta \left(\alpha \right)=\sum _{\alpha }{\alpha }_{\left(1\right)}\otimes {\alpha }_{\left(2\right)}\otimes {\alpha }_{\left(3\right)}\text{.}$$

The comultiplication $D\left(A\right)$ is determined by the formula

$$\Delta \left(a\alpha \right)=\sum _{a,\alpha }{a}_{\left(1\right)}{\alpha }_{\left(1\right)}\otimes a_{\left(2\right)}{\alpha }_{\left(2\right)},$$

where $\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}$ and $\Delta \left(\alpha \right)=\sum _{\alpha }{\alpha }_{\left(1\right)}\otimes {\alpha }_{\left(2\right)}\text{.}$

The relation between $D\left({𝔘}_h{𝔟}^{+}\right)$ and ${𝔘}_h𝔤$

With the definition of the quantum double in mind it is natural that we should define the quantum double of ${𝔘}_h{𝔟}^{+}$ to be the algebras

$$D\left({𝔘}_h{𝔟}^{+}\right)={\left({𝔘}_h{𝔟}^{+}\right)}^{*\text{coop}}\otimes {𝔘}_h{𝔟}^{+}\cong {𝔘}_h{𝔟}^{-}\otimes {𝔘}_h{𝔟}^{+}$$

with multiplication and comultiplcation given by the formulas in (7.2). The following theorem says that the quantum group ${𝔘}_h𝔤$ is almost the quantum double of ${𝔘}_h{𝔟}^{+},$ in other words, ${𝔘}_h𝔤$ is almost completely determined by pasting two copies of ${𝔘}_h{𝔟}^{+}$ together.

Let $\left(B_{ij}\right)=C^{-1}$ be the inverse of the Cartan matrix corresponding to $𝔤$ and, for each $1\le i\le r,$ define

$$H_i^{*}=\sum _{j=1}^r{B}_{ij}{H}_j\in {𝔘}_h𝔤\text{.}$$

1. There is a surjective homomorphism $\varphi :D\left({𝔘}_h{𝔟}^{+}\right)⟶{𝔘}_h𝔤$ determined by $$\begin{array}{cccc}\varphi :& D\left({𝔘}_h{𝔟}^{+}\right)& ⟶& {𝔘}_h𝔤\\ & X_i^{+}& ⟼& X_i^{+}\\ & H_i^{+}& ⟼& H_i\\ & X_i^{-}& ⟼& X_i^{-}\\ & H_i^{-}& ⟼& H_i^{*}\end{array}\text{and thus}\frac{D\left({𝔘}_h{𝔟}^{+}\right)}{\text{ker}\varphi }\cong {𝔘}_h𝔤\text{.}$$ (Recall (7.1) that we distinguish the elements $H_i$ which are in ${𝔘}_h{𝔟}^{+}$ from the elements $H_i$ which are in ${𝔘}_h{𝔟}^{-}$ by writing $H_i^{+}$ and $H_i^{-}$ respectively, instead of just $H_i$ in both cases.)
2. The ideal $\text{ker}\varphi$ is the ideal generated by the relations $$H_i^{-}-\left(\sum _{j=1}^r{B}_{ij}{H}_j^{+}\right),\text{where}1\le i\le r\text{.}$$

Using the $ℛ\text{-matrix}$ of $D\left({𝔘}_h{𝔟}^{+}\right)$ to get the $ℛ\text{-matrix}$ of ${𝔘}_h𝔤$

Recall (7.2) that the double $D\left({𝔘}_h{𝔟}^{+}\right)$ comes with a natural universal $ℛ\text{-matrix}$ given by

$$\stackrel{\sim }{ℛ}=\sum _i{b}_i\otimes b^i\text{.}$$

where the sum is over a basis $\left\{b_i\right\}$ of ${𝔘}_h{𝔟}^{+}$ and $\left\{b^i\right\}$ is the dual basis in ${𝔘}_h{𝔟}^{-}$ with respect to the form $⟨,⟩$ given in (2.1). We have used the notation $\stackrel{\sim }{ℛ}$ here to distinguish it from the element $ℛ$ in Theorem (3.2). The element $\stackrel{\sim }{ℛ}$ is not exactly in the tensor product $D\left({𝔘}_h{𝔟}^{+}\right)\otimes D\left({𝔘}_h{𝔟}^{+}\right)$ but if we make the tensor product just a tiny bit bigger by taking the $h\text{-adic}$ completion $D\left({𝔘}_h{𝔟}^{+}\right)⨶D\left({𝔘}_h{𝔟}^{+}\right)$ of $D\left({𝔘}_h{𝔟}^{+}\right)\otimes D\left({𝔘}_h{𝔟}^{+}\right)$ then we do have

$$\stackrel{\sim }{ℛ}\in D\left({𝔘}_h{𝔟}^{+}\right)⨶D\left({𝔘}_h{𝔟}^{+}\right)\text{.}$$

The image of $\stackrel{\sim }{ℛ}$ under the homomorphism

$$\begin{array}{cccc}\varphi \otimes \varphi :& D\left({𝔘}_h{𝔟}^{+}\right)⨶D\left({𝔘}_h{𝔟}^{+}\right)& ⟶& {𝔘}_h𝔤⨶{𝔘}_h𝔤\\ & \stackrel{\sim }{ℛ}& ⟼& ℛ\end{array}$$

coincides with the element $ℛ$ given in Theorem (3.2). This means that we actually get the element $ℛ$ in Theorem (3.2) for free by realising the quantum group as a quantum double (almost).

The quantum Serre relations occur naturally

In this section we will see that the most complicated of the definiing relations in the quantum group can be obtained in quite a natural way. More specifically, the ideal generated by them is the radical of a certain bilinear form.

Definition of the algebras $U_h{𝔟}^{+}$ and $U_h{𝔟}^{-}$

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 $U_h{𝔟}^{+}$ be the associative 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

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

with relations

$$[H_i,H_j]=0,\text{and}[H_i,X_j^{+}]={\alpha }_j\left(H_i\right)X_j^{+},\text{for all}1\le i,j\le r,$$

and define an algebra homomorphism ${\Delta }_h:U_h{𝔟}^{+}\to U_h{𝔟}^{+}⨶U_h{𝔟}^{+}$ by

$${\Delta }_h\left(H_i\right)=H_i\otimes 1+1\otimes H_i,\text{and}{\Delta }_h\left(X_i^{+}\right)=X_i^{+}\otimes e^{d_{i}h{H}_i}+1\otimes X_i^{+},$$

where $U_h{𝔟}^{+}⨶U_h{𝔟}^{+}$ denotes the $h\text{-adic}$ completion of the tensor product $U_h{𝔟}^{+}{\otimes }_{ℂ\left[\left[h\right]\right]}{U}_h{𝔟}^{+}\text{.}$

Let $U_h{𝔟}^{-}$ be the associative 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

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

with relations

$$[H_i,H_j]=0,\text{and}[H_i,X_j^{-}]=-{\alpha }_j\left(H_i\right)X_j^{-},\text{for all}1\le i,j\le r,$$

and define an algebra homomorphism ${\Delta }_h:U_h{𝔟}^{-}\to U_h{𝔟}^{-}⨶U_h{𝔟}^{-}$ by

$${\Delta }_h\left(H_i\right)=H_i\otimes 1+1\otimes H_i,\text{and}{\Delta }_h\left(X_i^{-}\right)=X_i^{-}\otimes 1+e^{-d_{i}h{H}_i}\otimes X_i^{-},$$

where $U_h{𝔟}^{-}⨶U_h{𝔟}^{-}$ denotes the $h\text{-adic}$ completion of the tensor product $U_h{𝔟}^{-}{\otimes }_{ℂ\left[\left[h\right]\right]}{U}_h{𝔟}^{-}\text{.}$

The difference between the algebras $U_h{𝔟}^{\pm }$ and the algebras ${𝔘}_h{𝔟}^{\pm }$

The algebras $U_h{𝔟}^{+}$ are much larger than the algebras ${𝔘}_h{𝔟}^{+}$ used in (2.1) since they have fewer relations between $X_i^{\pm }$ generators.

A pairing between $U_h{𝔟}^{+}$ and $U_h{𝔟}^{-}$

In exactly the same way that we had a pairing between ${𝔘}_h{𝔟}^{+}$ and ${𝔘}_h{𝔟}^{-}$ in (2.1), there is a unique $ℂ\left[\left[h\right]\right]\text{-bilinear}$ pairing

$$⟨,⟩:U_h{𝔟}^{-}\times U_h{𝔟}^{+}⟶ℂ\left[\left[h\right]\right]\text{which satisfies}$$

1. $⟨1,1⟩=1,$
2. $⟨H_i,H_j⟩=\frac{{\alpha }_j\left(H_i\right)}{d_j},$
3. $⟨X_i^{-},X_j^{+}⟩={\delta }_{ij}\frac{h}{e^{d_{i}h}-e^{-d_{i}h}},$
4. $⟨ab,c⟩=⟨a\otimes b,{\Delta }_h\left(c\right)⟩,$ for all $a,b\in U_h{𝔟}^{-}$ and $c\in U_h{𝔟}^{+},$
5. $⟨a,bc⟩=⟨{\Delta }_h^{\text{op}}\left(a\right),b\otimes c⟩,$ for all $a\in U_h{𝔟}^{-}$ and $b,c\in U_h{𝔟}^{+}\text{.}$

The radical of $⟨,⟩$ is generated by the quantum Serre relations

Let ${𝔯}^{-}$ and ${𝔯}^{+}$ be the left and right radicals, respectively, of the form $⟨,⟩$ defined in (8.3), i.e.

$$\begin{array}{ccc}{𝔯}^{-}& =& \{a\in U_h{𝔟}^{-}\mid ⟨a,b⟩=0\text{for all}b\in U_h{𝔟}^{+}\},\text{and}\\ {𝔯}^{+}& =& \{b\in U_h{𝔟}^{+}\mid ⟨a,b⟩=0\text{for all}a\in U_h{𝔟}^{-}\}\text{.}\end{array}$$

The sets ${𝔯}^{-}$ and ${𝔯}^{+}$ are the ideals of $U_h{𝔟}^{-}$ and $U_h{𝔟}^{+}$ generated by the elements

$$\sum _{s+t=1-{\alpha }_j\left(H_i\right)}{(-1)}^s{\left[\genfrac{}{}{0ex}{}{1-{\alpha }_j\left(H_i\right)}{s}\right]}_{e^{d_{i}h}}{\left(X_i^{-}\right)}^s{X}_j^{-}{\left(X_i^{-}\right)}^t,\text{for}i\ne j,$$

and

$$\sum _{s+t=1-{\alpha }_j\left(H_i\right)}{(-1)}^s{\left[\genfrac{}{}{0ex}{}{1-{\alpha }_j\left(H_i\right)}{s}\right]}_{e^{d_{i}h}}{\left(X_i^{+}\right)}^s{X}_j^{+}{\left(X_i^{+}\right)}^t,\text{for}i\ne j,$$

respectively.

It follows from this theorem that the quantum group ${𝔘}_h𝔤$ is determined by the algebras $U_h{𝔟}^{+},$ $U_h{𝔟}^{-}$ and the form $⟨,⟩\text{.}$ A construction of the quantum group along these lines would be very similar to the standard construction of Kac-Moody Lie algebras (see [Kac1983] §1.3).

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.

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