Quantum Groups

Last update: 26 July 2013

Quantum groups

For a symbol $q$ use the notation

$[n]q= qn-q-n q-q-1 =qn-1+ qn-3+… +q-(n-3) +q-(n-1), for n∈ℤ, [n]q!= [n]q [n-1]q… [2]q [1]q, for n∈ℤn≥0, [mn]q= [m]q [m-1]q … [m-n+1]q [n]q! , for m∈ℤ and n∈ℤ≥0, (1.1)$

Let $G$ be a complex reductive algebraic group, $B$ a Borel subgroup and $T\subseteq B$ a maximal torus of $G\text{.}$ Let

$𝔥ℤ=Hom(ℂ×,T) and𝔥ℤ*= Hom(T,ℂ×),$

and let

${α1,…,αn} ⊆𝔥ℤ*and { α1∨,…, αn∨ } ⊆𝔥ℤ$

the simple roots and coroots corresponding the Borel subgroup $B\text{.}$ Let $A$ be the corresponding $n×n$ Cartan matrix and let ${d}_{1},\dots ,{d}_{n}\in {ℤ}_{>0}$ be minimal such that $\text{diag}\left({d}_{1},\dots ,{d}_{n}\right)·A$ is symmetric and let ${q}_{i}={q}^{{d}_{i}},$

$A= (⟨αi,αj∨⟩) 1≤i,j≤n anddi= 2⟨αi,αi⟩ andqi=qdi.$

The quantum group ${U}_{q}\left(G\right)$ is the Hopf algebra over $ℚ\left(q\right)$ generated by

$Ei,Fi,1≤ i≤n,and Kλ,λ∈𝔥ℤ,$

with relations

$KλKμ= Kλ+μ, KλEjK-λ= q⟨λ,αj⟩ Ej,KλFj K-λ= q-⟨λ,αj⟩ Fj, EiFi-FiEi= δij Kαi∨-Kαi∨-1 qi-qi-1 , ∑s=01-⟨αi,αj∨⟩ (-1)s [1-⟨αi,αj∨⟩s]qi Ei1-⟨αiαj∨⟩-s EjEis=0,for i≠j, ∑s=01-⟨αi,αj∨⟩ (-1)s [1-⟨αi,αj∨⟩s]qi Fi1-⟨αiαj∨⟩-s FjFis=0,for i≠j,$

with

$Δ(Ei)= Ei⊗Kαi∨+1 ⊗Ei, Δ(Fi)= Fi⊗1+ Kαi∨-1⊗ Fi, Δ(Kλ)= Kλ⊗Kλ, S(Ei)=-Ei Kαi∨-1, S(Fi)=- Kαi∨Fi, S(Kαi∨)= Kαi∨-1, ε(Ei)=0, ε(Fi)=0, ε(Kλ)=1,$

(see [CPr1994, Section 9.1]). There is a triangular decomposition

$Uq>0 is the subalgebra of Uq generated by Ei, Uq= Uq<0 Uq0 Uq>0, where Uq0 is the subalgebra of Uq generated by Kλ, Uq<0 is the subalgebra of Uq generated by Fi,$

(see [CPr1994 ,Proposition 9.1.3]). For $1\le i\le n$ and $a,r\in ℤ,r\ge 0,$ let

$Ki=Kαi∨, (Ei)(r)= Eir[r]qi! , (Fi)(r)= Fir[r]qi! , and let (1.2) [qicKik]qi= ( qicKi- qi-c Ki-1 qi-qi-1 ) ( qic-1Ki- qi-(c-1) Ki-1 qi2-qi-2 ) … ( qic-(k-1) Ki- qi-(c-(k-1)) Ki-1 qik-qi-k ) ,$

for $i\in \left\{1,\dots ,m\right\},$ $c\in ℤ$ and $k\in {ℤ}_{\ge 0}\text{.}$ The identity

$Ei(r) Fi(s)= ∑k=0min(r,s) Fi(s-k) [ qi-(s-k+r-k)Ki k ] qi Ei(r-k) (1.3)$

is proved first for $r=1$ by induction on $s,$ and then by induction on $r$ for a fixed $s$ (see Proposition 1.1 and [Lus1993, Corollary 3.1.9]).

Let $𝔸=ℤ\left[q,{q}^{-1}\right]\text{.}$ The restricted integral form of ${U}_{q}$ is the $𝔸\text{-Hopf}$ subalgebra ${U}_{𝔸}$ generated by

$(Ei)(r), (Fi)(r), andKλ,for 1≤i≤n and r∈ ℤ≥0 and λ∈ 𝔥ℤ$

(see [CPr1994, Definition-Proposition 9.3.1]). Intersecting the triangular decomposition of ${U}_{q}$ with ${U}_{𝔸},$ the $𝔸\text{-algebra}$ ${U}_{𝔸}$ has a triangular decomposition

$U𝔸>0 is the subalgebra of U𝔸 generated by (Ei)(r), U𝔸= U𝔸<0 U𝔸0 U𝔸>0, where U𝔸0 is the subalgebra of U𝔸 generated by Kλ and [Ki;0j]qi, U𝔸<0 is the subalgebra of U𝔸 generated by (Fi)(r),$

(see [CPr1994, Proposition 9.3.3]).

As discussed in [Lus1990-2, Section 6], ${U}_{𝔸}\left(G\right)$ is an integral form of ${U}_{q}\left(G\right)$ in the sense that

$U𝔸(G)⊗𝔸 ℚ(q)=Uq(G)$

$\text{(}q,$ as an indeterminate, is not a root of unity). As in [CPr1994, Section 9.3A],

$for ε∈ℂ× defineUε(G) =U𝔸(G) ⊗𝔸ℂ (1.4)$

where $ℂ$ is given an $𝔸\text{-algebra}$ via the ring homomorphism which maps $q$ to $\epsilon \text{.}$ The remarks just before [CPr1994, Proposition 9.3.5] indicate that if $\epsilon$ is not a root of unity then ${U}_{\epsilon }\left(G\right)$ is the associative algebra generated by ${E}_{i},$ ${F}_{i}$ and ${K}_{\lambda }$ with relations as in the definition of ${U}_{q}\left(G\right)$ except with $q$ replaced by $\epsilon \text{.}$