The quantum groups GL2, SL2 and PGL2

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

Last update: 26 July 2013

The quantum groups GL2, SL2 and PGL2

The quantum group Uq(GL2) is the algebra given by generators E,F,K1±1,K2±1 and relations

K1-1K1= K1K1-1= 1, K2-1K2= K2K2-1= 1, K1K2= K2K1, (1.1)


K1EK1-1= qε1-ε2,ε1 E=qE,K2EK2-1= qε1-ε2,ε2 E=q-1E, (1.2) K1FK1-1= q-ε1-ε2,ε1 F=q-1F, K2FK2-1= q-ε1-ε2,ε2 F=qF, (1.3) EF-FE= K1K2-1- K1-1K2 q-q-1 . (1.4)

The coproduct is given by

Δ(Ki±1) =Ki±1 Ki±1,ε (Ki±1)=1 S(Ki±1)= Ki1 (fori=1,2) Δ(E)= E1+K1K2-1 E,ε(E)=0S (E)=-K1-1 K2E, Δ(F)= FK1-1K2+1 F,ε(F)=0S (F)=-FK1 K2-1,

The quantum group Uq(SL2) is the subalgebra of Uq(GL2) generated by E,F and K=K1K2-1. Alternatively, the quantum group Uq(SL2) is the algebra generated by E,F and K±1 with relations

KK-1=K-1K=1, KEK-1=q2E, KFK-1=q-2F, EF-FE=K-K-1q-q-1.

The quantum group Uq(PGL2) is Uq(GL2) with the additional relation K1K2=1.

In summary, there are Hopf algebra homomorphisms

Uq(SL2) Uq(GL2) π Uq(PGL2) = Uq(GL2)K1K2-1 EEE FFF KK1K2-1 K1K21

The short exact sequences

1SL2GL2 det×1and 1×GL2 πPGL21


𝔥PGL* 𝔥GL* 𝔥SL*= 𝔥GL*(ε1+ε2) kε1-kε2 kε1-kε2 fork, aε1+bε2 [(a-b)ε1]


[𝔥PGL*]= [Y±1], [𝔥GL*]= [Y1±1,Y2±1], [𝔥SL*]= [Y1±1],


[𝔥PGL*] [𝔥GL*] [𝔥SL*]= [𝔥GL*] Y2=Y1-1 Y Y1Y2-1 Y1aY2b Y1a-b

The dominant weights

(𝔥G*)+= { aε1+bε2| a,bwithab }

are the elements of 𝔥G* on the positive side of the hyperplane

𝔥ε1-ε2= { aε1+bε2| a,bwitha-b=0 } .

and the Weyl group W0={1,s1} has s12=1, with s1 reflection in the hyperplane 𝔥ε1-ε2. Then

Rep(PGL2) [𝔥PGL*]W0, Rep(GL2) [𝔥GL*]w0 Rep(SL2) [𝔥SL*]w0

and the representation theories of PGL2,GL2 and SL2 are related by

π*:Rep (PGL2) Rep(GL2)and Rep(GL2)= {deti|i} Rep(SL2)

so that

Rep(PGL2) π* Rep(GL2) res Rep(SL2)= Rep(GL2)det L(kα) L(k(ε1-ε2)) fork0, L(aε1+bε2) L((a-b)ε1) forab


ε2 ε1

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