Quantum Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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), forn, [n]q!= [n]q [n-1]q [2]q [1]q, fornn0, [mn]q= [m]q [m-1]q [m-n+1]q [n]q! , formand n0, (1.1)

Let G be a complex reductive algebraic group, B a Borel subgroup and TB a maximal torus of G. Let

𝔥=Hom(×,T) and𝔥*= Hom(T,×),

and let

{α1,,αn} 𝔥*and { α1,, αn } 𝔥

the simple roots and coroots corresponding the Borel subgroup B. Let A be the corresponding n×n Cartan matrix and let d1,,dn>0 be minimal such that diag(d1,,dn)·A is symmetric and let qi=qdi,

A= (αi,αj) 1i,jn anddi= 2αi,αi andqi=qdi.

The quantum group Uq(G) is the Hopf algebra over (q) generated by

Ei,Fi,1 in,and Kλ,λ𝔥,

with relations

KλKμ= Kλ+μ, KλEjK-λ= qλ,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,αjs]qi Ei1-αiαj-s EjEis=0,for ij, s=01-αi,αj (-1)s [1-αi,αjs]qi Fi1-αiαj-s FjFis=0,for ij,


Δ(Ei)= EiKαi+1 Ei, Δ(Fi)= Fi1+ Kαi-1 Fi, Δ(Kλ)= KλKλ, S(Ei)=-Ei Kαi-1, S(Fi)=- KαiFi, 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 ofUq generated byEi, Uq= Uq<0 Uq0 Uq>0, where Uq0 is the subalgebra ofUq generated byKλ, Uq<0 is the subalgebra ofUq generated byFi,

(see [CPr1994 ,Proposition 9.1.3]). For 1in and a,r,r0, 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{1,,m}, c and k0. 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=[q,q-1]. The restricted integral form of Uq is the 𝔸-Hopf subalgebra U𝔸 generated by

(Ei)(r), (Fi)(r), andKλ,for 1inandr 0andλ 𝔥

(see [CPr1994, Definition-Proposition 9.3.1]). Intersecting the triangular decomposition of Uq with U𝔸, the 𝔸-algebra U𝔸 has a triangular decomposition

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

(see [CPr1994, Proposition 9.3.3]).

As discussed in [Lus1990-2, Section 6], U𝔸(G) is an integral form of Uq(G) in the sense that

U𝔸(G)𝔸 (q)=Uq(G)

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

forε× defineUε(G) =U𝔸(G) 𝔸 (1.4)

where is given an 𝔸-algebra via the ring homomorphism which maps q to ε. The remarks just before [CPr1994, Proposition 9.3.5] indicate that if ε is not a root of unity then Uε(G) is the associative algebra generated by Ei, Fi and Kλ with relations as in the definition of Uq(G) except with q replaced by ε.

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