## VII. Properties of quantum groups

Last update: 2 November 2012

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_{h}𝔤$ be the Drinfel'd-Jimbo quantum group corresponding to $𝔤$ that was defined in V (1.3). We shall often use the presentation of ${𝔘}_{h}𝔤$ 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 ${𝔘}_{h}𝔤$ given in section 6 appears in detail in [Jan1995] 8.18-8.30. The statement that ${𝔘}_{h}𝔤$ 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 of ${𝔘}_{h}𝔤$

The triangular decomposition of the quantum group ${𝔘}_{h}𝔤$ 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 ${𝔘}_{h}𝔤$ be the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). Define

$𝔘h𝔫- = subalgebra of𝔘h𝔤 generated by X1-,X2-,…, Xr-, 𝔘h𝔥 = subalgebra of𝔘h𝔤 generated by H1,H2,…,Hr, 𝔘h𝔫+ = subalgebra of𝔘h𝔤 generated by X1+,X2+,…, Xr+.$

The map

$𝔘h𝔫-⊗ 𝔘h𝔥⊗𝔘h𝔫+ ⟶ 𝔘h𝔤 u-⊗u0⊗u+ ⟼ u-u0u+$

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 ${𝔘}_{h}𝔤$ 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 ${𝔘}_{h}𝔤$ 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

$Q+=∑iℕαi,whereℕ=ℤ≥0.$

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

$(𝔘h𝔫+)ν = span- { Xi1+… Xip+ ∣ Xi1+… Xip+ has νj-factors of type Xj+ } (𝔘h𝔫-)ν = span- { Xi1-… Xip- ∣ Xi1-… Xip- has νj-factors of type Xj- } .$

Then

$𝔘h𝔫-= ⨁ν∈Q+ (𝔘h𝔫-)ν and 𝔘h𝔫+= ⨁ν∈Q+ (𝔘h𝔫+)ν,$

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 ${𝔘}_{h}𝔤\text{.}$ The extended pairing is an analogue of the Killing form on $𝔤$ in two ways:

1. it is an ad-invariant form on ${𝔘}_{h}𝔤,$ and
2. upon restriction to $𝔤$ it coincides $\phantom{\rule{1em}{0ex}}\left(\text{mod}\phantom{\rule{0.2em}{0ex}}h\right)$ 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 ${𝔘}_{h}𝔤$ be the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). Define

$𝔘h𝔟- = subalgebra of𝔘h𝔤 generated by X1-,X2-,…, Xr-and H1,…,Hr, 𝔘h𝔟+ = subalgebra of𝔘h𝔤 generated by X1+,X2+,…, Xr+and H1,…,Hr.$

1. There is a unique $ℂ\left[\left[h\right]\right]\text{-bilinear}$ pairing $⟨,⟩: 𝔘h𝔟-× 𝔘h𝔟+⟶ℂ [[h]] which satisfies$
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. $⟨ab,c⟩=⟨a\otimes b,{\Delta }_{h}\left(c\right)⟩,\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}a,b\in {𝔘}_{h}{𝔟}^{-}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\in {𝔘}_{h}{𝔟}^{+},$
5. $⟨a,bc⟩=⟨{\Delta }_{h}^{\text{op}}\left(a\right),b\otimes c⟩,\phantom{\rule{1em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}a\in {𝔘}_{h}{𝔟}^{-}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b,c\in {𝔘}_{h}{𝔟}^{+}\text{.}$
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 $\mu ,\nu \in {Q}^{+}\text{.}$ $Ifμ≠ν then ⟨ (𝔘h𝔟-)μ, (𝔘h𝔟+)ν ⟩ =0.$
2. Let $\nu \in {Q}^{+}\text{.}$ The restriction of the pairing $⟨,⟩$ to ${\left({𝔘}_{h}{𝔫}^{-}\right)}_{\nu }×{\left({𝔘}_{h}{𝔫}^{+}\right)}_{\nu }$ is a nondegenerate pairing $⟨,⟩: (𝔘h𝔫-)ν× (𝔘h𝔫+)ν →ℂ[[h]].$

If $\theta$ is the Cartan involution of ${𝔘}_{h}𝔤$ as given in V (1.6) and ${S}_{h}$ is the antipode of ${𝔘}_{h}𝔤$ then

$⟨ θ(u+), θ(u-) ⟩ =⟨u-,u+⟩ and ⟨ Sh(u-), Sh(u+) ⟩ =⟨u-,u+⟩,$

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

### Extending the pairing to an ad-invariant pairing on ${𝔘}_{h}𝔤$

The triangular decomposition (1.1) of ${𝔘}_{h}𝔤$ says that ${𝔘}_{h}𝔤\cong {𝔘}_{h}{𝔫}^{-}\otimes {𝔘}_{h}𝔥\otimes {𝔘}_{h}{𝔫}^{+}$ and that

$every elementu∈𝔘h𝔤 can be written in the formu-u0 u+,$

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

$⟨,⟩: 𝔘h𝔤×𝔘h𝔤⟶ ℂ[[h]] defined by the formula ⟨ u1-u10 u1+,u2- u20u2+ ⟩ = ⟨ u1-,Sh (u20u2+) ⟩ ⟨ u2-,Sh-1 (u10u1+) ⟩ ,$

for all ${u}_{1}^{-},{u}_{2}^{-}\in {𝔘}_{h}{𝔫}^{-},$ ${u}_{1}^{0},{u}_{2}^{0}\in {𝔘}_{h}𝔥,$ and ${u}_{1}^{+},{u}_{2}^{+}\in {𝔘}_{h}{𝔫}^{+},$ where ${S}_{h}$ is the antipode of ${𝔘}_{h}𝔤\text{.}$ Then

$⟨ adu(v1),v2 ⟩ = ⟨ v1,adSh(u) (v2) ⟩ ,for allu,v1, v2∈𝔘h𝔤,$

This formula says that the extended pairing $⟨,⟩$ is an ad-invariant pairing as defined in I (5.5). The pairing $⟨,⟩:\phantom{\rule{0.2em}{0ex}}{𝔘}_{h}𝔤\to {𝔘}_{h}𝔤ℂ\left[\left[h\right]\right]$ is not symmetric, see I (5.5).

### Duality between matrix coefficients for representations and ${U}_{q}𝔤\text{.}$

Let $𝔤$ be finite dimensional complex simple Lie algebra and let ${U}_{q}𝔤$ 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 ${U}_{q}𝔤$ 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 $u\in {U}_{q}𝔤$ such that

$n*(vm)= ⟨v,u⟩,for all v∈Uq𝔤,$

where $⟨,⟩$ is the bilinear form on ${U}_{q}𝔤$ given by (2.1) after making the substitutions in V (2.2).

The function

$cm,n*: Uq𝔤→ℂ(q) defined bycm,n* (v)= ⟨n*,vm⟩$

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

## 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 $\left({𝔘}_{h}𝔤,ℛ\right)$ 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 ${𝔘}_{h}𝔤$ 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

$t0=∑i=1r H∼i⊗H∼i.$

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 ${𝔘}_{h}𝔤\otimes {𝔘}_{h}𝔤$ so we have to make the tensor product just a tiny bit bigger. To do this we let ${𝔘}_{h}𝔤⨶{𝔘}_{h}𝔤$ denote the $h\text{-adic}$ completion of the tensor product ${𝔘}_{h}𝔤\otimes {𝔘}_{h}𝔤,$ see III §1.

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

$ℛΔh(a) ℛ-1=Δhop (a),for all a∈𝔘h𝔤,and ℛhas the formℛ= ∑ν∈Q+ exp { h ( t0+12 ( Hν⊗1-1⊗Hν ) ) } Pν,where$ $Pν∈ (𝔘h𝔫-)ν⊗ (𝔘h𝔫+)ν, Hν=∑i=1r νiHi,if ν=∑iνiαi, Pνis a polynomial in Xi+⊗1and 1⊗Xi-,1≤i ≤r,with coefficients in ℂ[[h]], such that the smallest power of h inPν with nonzero coefficients ishnν .$

### Properties of the $ℛ\text{-matrix}$

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

$ℛ12=∑ai⊗ bi⊗1,ℛ13 =∑ai⊗1⊗ bi, andℛ23=∑1⊗ ai⊗bi, and letℛ21=∑ bi⊗ai.$

Let $\sigma :\phantom{\rule{0.2em}{0ex}}{𝔘}_{h}𝔤\to {𝔘}_{h}𝔤$ be the $ℂ\text{-algebra}$ automorphism of ${𝔘}_{h}𝔤$ given by $\sigma \left(h\right)=-h,$ $\sigma \left({X}_{i}^{±}\right)={X}_{i}^{±},$ and $\sigma \left({H}_{i}\right)={H}_{i}\text{.}$ With these notations we have

$(Δh⊗id) (ℛ)=ℛ13 ℛ23,and (id⊗Δh) (ℛ)=ℛ13 ℛ12, (εh⊗id)(ℛ) =1=(id⊗εh) (ℛ), (Sh⊗id)(ℛ)= (id⊗Sh-1) (ℛ)=ℛ-1 and(Sh⊗Sh) (ℛ)=ℛ, (θ⊗θ)(ℛ)= ℛ21and (σ⊗σ)(ℛ)= ℛ-1.$

## An analogue of the Casimir element

### Definition of the element $u$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_{h}𝔤$ be the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). The antipode ${S}_{h}:\phantom{\rule{0.2em}{0ex}}{𝔘}_{h}𝔤\to {𝔘}_{h}𝔤$ is an antiautomorphism of ${𝔘}_{h}𝔤,$ see I (2.1). This means that the map ${S}_{h}^{2}:\phantom{\rule{0.2em}{0ex}}{𝔘}_{h}𝔤\to {𝔘}_{h}𝔤$ is an automorphism of ${𝔘}_{𝔤}\text{.}$ The following theorem says that this automorphism is inner!

Let $ℛ\in {𝔘}_{h}𝔤⨶{𝔘}_{h}𝔤$ be the universal $ℛ\text{-matrix}$ of ${𝔘}_{h}𝔤$ 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

$uau-1=Sh2 (a),for alla ∈𝔘h𝔤.$

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

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

$Δh(u)= (ℛ21ℛ12) -1 (u⊗u),Sh (u)=u,and εh(u)=1,$

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

$u-1=∑Sh-1 (dj)cj,where ℛ-1=∑cj⊗ dj.$

### 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 ${𝔘}_{h}𝔤$ shows that

$ehρ∼a e-hρ∼= Sh2(a), for alla∈𝔘h𝔤.$

It follows that

$the elemente-hρ∼u =ue-hρ∼ is a central element in𝔘h𝔤.$

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

$ρ=12∑α>0 α,$

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

$e-hρ∼u acts onL(λ) by the constant q-(λ+ρ,λ+ρ)+(ρ,ρ) ,$

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

$(e-hρ∼u)2 =uSh(u).$

## 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

$ϕ(Xi±)= Xj±,andϕ (Hi)=Hj, wherew0siw0-1 =sj,for1≤i≤r,$

extends to an automorphism of ${𝔘}_{h}𝔤\text{.}$ Let $\stackrel{\sim }{\theta }$ be the anti-automorphism of ${𝔘}_{h}𝔤$ defined by $\stackrel{\sim }{\theta }\left({X}_{i}^{±}\right)={X}_{i}^{\mp }$ 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 ${𝔘}_{h}𝔤$ as given in V (1.6). There are both anti-automorphisms of ${𝔘}_{h}𝔤\text{.}$ The composition.

$(Sh∘θ∼∘ϕ) :𝔘h𝔤→𝔘h𝔤$

is an automorphism of ${𝔘}_{h}𝔤\text{.}$ 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 ${𝔘}_{h}𝔤$ be the corresponding quantum group as presented in V (1.6). Let $q={e}^{h}$ and for each $1\le i\le r$ let

$Ei(r)= (Xi+)r [r]qdi! , Fi(r)= (Xi+)r [r]qdi! andKi= ehdiHi,$

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

$Ti=∑a,b,c≥0 (-1)b q b-ac+(c+a-b) (c-a) Ei(a) Fi(b) Ei(c) Kic-a,$

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

1. The elements ${T}_{i}$ satisfy the relations $TiTjTiTj… ⏟mijfactors = TjTiTjTi… ⏟mijfactors fori≠j,$ 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 $Tw0=Ti1 …TiN.$ Then ${T}_{{w}_{0}}$ is invertible and $Tw0aTw0-1= (Sh⊗θ∼⊗ϕ) (a),for alla∈ 𝔘h𝔤.$

### Properties of the element ${T}_{{w}_{0}}$

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

$σ(Tw0)Tw0 =uandTw0-1 =σ∼(Tw0).$

The relationship between the element ${T}_{{w}_{0}}$ and the Hopf algebra structure of ${𝔘}_{h}𝔤$ is given by the formulas

$Δh(Tw0)= ℛ12-1 (Tw0⊗Tw0)= (Tw0⊗Tw0) ℛ21-1, Sh(Tw0)= Tw0ehρ∼, andεh (Tw0)=1,$

where ${ℛ}_{12}=ℛ=\sum {a}_{i}\otimes {b}_{i}$ is the universal $ℛ\text{-matrix}$ of ${𝔘}_{h}𝔤$ given in (3.2), and ${ℛ}_{21}=\sum {b}_{i}\otimes {a}_{i}\text{.}$

## The Poincaré-Birkhoff-Witt bais of ${𝔘}_{h}𝔤$

### Root vectors in ${𝔘}_{h}𝔤$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_{h}𝔤$ be the corresponding quantum group as presented in V (1.6). Let ${T}_{i}$ be the elements of ${𝔘}_{h}𝔤$ given in (5.2). Define an automorphism ${\tau }_{i}:\phantom{\rule{0.2em}{0ex}}{𝔘}_{h}𝔤\to {𝔘}_{h}𝔤$ by

$τi(u)=Ti uTi-1, for allu∈𝔘h𝔤,$

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

$β1=αi1, β2=si1 (αi2),…, βN=si1 si2…siN-1 (αiN).$

The elements ${\beta }_{1},\dots ,{\beta }_{N}$ are the positive roots $𝔤\text{.}$ Define elements of ${𝔘}_{h}𝔤$ by

$Xβ1±= Xi1±, Xβ2±= τi1(Xi2±) ,…, XβN±=τi1 τi2…τiN-1 (XiN±).$

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}_{h}𝔥,$ and ${𝔘}_{h}{𝔫}^{+}$

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

Let ${X}_{{\beta }_{1}}^{±},\dots ,{X}_{{\beta }_{N}}^{±}$ be the elements of ${𝔘}_{h}𝔤$ defined in (6.1). Then

${ (Xβ1+)p1 (Xβ2+)p2 … (XβN+)pN ∣ p1,…,pN ∈ℤ≥0 } is a basis of 𝔘h𝔫+, { (Xβ1-)n1 (Xβ2-)n2 … (XβN-)nN ∣ n1,…,nN ∈ℤ≥0 } is a basis of 𝔘h𝔫-, { H1s1 H2s2 … H1sr ∣ s1,…,sN ∈ℤ≥0 } is a basis of 𝔘h𝔥.$

### 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 ${X}_{{\beta }_{j}}^{±},$ $1\le j\le N,$ be the elements defined in (6.1). Let ${p}_{1},\dots ,{p}_{N},$ ${n}_{1},\dots ,{n}_{N}\in {ℤ}_{\ge 0}\text{.}$ Then

$⟨ (Xβ1-)n1 (Xβ2-)n2 … (XβN-)nN , (Xβ1+)p1 (Xβ2+)p2 … (XβN+)pN ⟩ =∏j=1N δnj,pj ⟨ (Xij-)nj , (Xij+)njs ⟩ ,$

where ${\delta }_{{n}_{j},{p}_{j}}$ is the Kronecker delta.

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

$⟨ (Xi-)n , (Xi+)n ⟩ =(-1)n q -din(n-1) /2 [n]qdi! (qdi-q-di) n ,whereq=eh.$

## 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 ${𝔘}_{h}𝔤$ be the corresponding quantum group as presented in V (1.6). Define

$𝔘h𝔟- = subalgebra of𝔘h𝔤 generated by X1-, X2-,…, Xr- and H1,…,Hr, 𝔘h𝔟+ = subalgebra of𝔘h𝔤 generated by X1+, X2+,…, Xr+ and H1,…,Hr,$

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

$⟨x1x2,y⟩= ⟨ x1⊗x2, Δh(y) ⟩ and ⟨x,y1y2⟩ = ⟨ Δop(x), y1⊗y2 ⟩$

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),

$𝔘h𝔟-≃ (𝔘h𝔟+)*coop as Hopf algebras,$

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

$D(A)= { aα∣a∈A, α∈A*coop } ≅A⊗A*coop$

with multiplication determined by the formulas

$αa = ∑α,a ⟨ α(1), S-1 (a(1)) ⟩ ⟨ α(3), a(3) ⟩ a(2)α(2), and aα = ∑α,a ⟨ α(1), a(1) ⟩ ⟨ α(3), S-1 (a(3)) ⟩ α(2) a(2),$

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

$(Δ⊗id)∘ Δ(a)=∑a a(1)⊗ a(2)⊗ a(3),and (Δ⊗id)∘ Δ(α)= ∑αα(1)⊗ α(2)⊗α(3).$

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

$Δ(aα)= ∑a,α a(1) α(1)⊗ a(2) α(2),$

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 ${𝔘}_{h}𝔤$

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

$D(𝔘h𝔟+)= (𝔘h𝔟+)*coop ⊗𝔘h𝔟+≅ 𝔘h𝔟-⊗ 𝔘h𝔟+$

with multiplication and comultiplcation given by the formulas in (7.2). The following theorem says that the quantum group ${𝔘}_{h}𝔤$ is almost the quantum double of ${𝔘}_{h}{𝔟}^{+},$ in other words, ${𝔘}_{h}𝔤$ 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

$Hi*=∑j=1r BijHj∈ 𝔘h𝔤.$
1. There is a surjective homomorphism $\varphi :\phantom{\rule{0.2em}{0ex}}D\left({𝔘}_{h}{𝔟}^{+}\right)⟶{𝔘}_{h}𝔤$ determined by $ϕ: D(𝔘h𝔟+) ⟶ 𝔘h𝔤 Xi+ ⟼ Xi+ Hi+ ⟼ Hi Xi- ⟼ Xi- Hi- ⟼ Hi* and thus D(𝔘h𝔟+) kerϕ ≅𝔘h𝔤.$ (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 $\text{ker}\phantom{\rule{0.2em}{0ex}}\varphi$ is the ideal generated by the relations $Hi-- ( ∑j=1r BijHj+ ) ,where 1≤i≤r.$

### Using the $ℛ\text{-matrix}$ of $D\left({𝔘}_{h}{𝔟}^{+}\right)$ to get the $ℛ\text{-matrix}$ of ${𝔘}_{h}𝔤$

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

$ℛ∼=∑ibi ⊗bi.$

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

$ℛ∼∈D(𝔘h𝔟+) ⨶D(𝔘h𝔟+).$

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

$ϕ⊗ϕ: D(𝔘h𝔟+)⨶ D(𝔘h𝔟+) ⟶ 𝔘h𝔤⨶𝔘h𝔤 ℛ∼⟼ℛ$

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

$H1,H2,…,Hr, X1r,X2r, …,Xr+,$

with relations

$[Hi,Hj]=0, and [Hi,Xj+]= αj(Hi) Xj+,for all 1≤i,j≤r,$

and define an algebra homomorphism ${\Delta }_{h}:\phantom{\rule{0.2em}{0ex}}{U}_{h}{𝔟}^{+}\to {U}_{h}{𝔟}^{+}⨶{U}_{h}{𝔟}^{+}$ by

$Δh(Hi)= Hi⊗1+1⊗Hi, andΔh (Xi+)= Xi+⊗ edihHi+1⊗ Xi+,$

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

$X1-,X2-,…, Xr-,H1, H2,…,Hr,$

with relations

$[Hi,Hj]=0, and[Hi,Xj-] =-αj(Hi) Xj-,for all 1≤i,j≤r,$

and define an algebra homomorphism ${\Delta }_{h}:\phantom{\rule{0.2em}{0ex}}{U}_{h}{𝔟}^{-}\to {U}_{h}{𝔟}^{-}⨶{U}_{h}{𝔟}^{-}$ by

$Δh(Hi) =Hi⊗1+1⊗Hi, andΔh (Xi-)= Xi-⊗1+ e-dihHi ⊗Xi-,$

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 ${U}_{h}{𝔟}^{±}$ and the algebras ${𝔘}_{h}{𝔟}^{±}$

The algebras ${U}_{h}{𝔟}^{+}$ are much larger than the algebras ${𝔘}_{h}{𝔟}^{+}$ used in (2.1) since they have fewer relations between ${X}_{i}^{±}$ 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

$⟨,⟩: Uh𝔟-× Uh𝔟+ ⟶ℂ[[h]] which satisfies$
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. $⟨ab,c⟩=⟨a\otimes b,{\Delta }_{h}\left(c\right)⟩,\phantom{\rule{1em}{0ex}}$ for all $a,b\in {U}_{h}{𝔟}^{-}$ and $c\in {U}_{h}{𝔟}^{+},$
5. $⟨a,bc⟩=⟨{\Delta }_{h}^{\text{op}}\left(a\right),b\otimes c⟩,\phantom{\rule{1em}{0ex}}$ 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.

$𝔯- = { a∈Uh𝔟- ∣ ⟨a,b⟩=0 for all b∈Uh𝔟+ } ,and 𝔯+ = { b∈Uh𝔟+ ∣ ⟨a,b⟩=0 for all a∈Uh𝔟- } .$

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

$∑s+t=1-αj(Hi) (-1)s [ 1-αj(Hi) s ] edih (Xi-)s Xj- (Xi-)t, fori≠j,$

and

$∑s+t=1-αj(Hi) (-1)s [ 1-αj(Hi) s ] edih (Xi+)s Xj+ (Xi+)t, fori≠j,$

respectively.

It follows from this theorem that the quantum group ${𝔘}_{h}𝔤$ 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.