## The quantum groups ${GL}_{2},$${SL}_{2}$ and ${PGL}_{2}$

Last update: 26 July 2013

## The quantum groups ${GL}_{2},$${SL}_{2}$ and ${PGL}_{2}$

The quantum group ${U}_{q}\left({GL}_{2}\right)$ is the algebra given by generators $E,F,{K}_{1}^{±1},{K}_{2}^{±1}$ and relations

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

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

The coproduct is given by

$Δ(Ki±1) =Ki±1⊗ Ki±1,ε (Ki±1)=1 S(Ki±1)= Ki∓1 (for i=1,2) Δ(E)= E⊗1+K1K2-1⊗ E,ε(E)=0S (E)=-K1-1 K2E, Δ(F)= F⊗K1-1K2+1⊗ F,ε(F)=0S (F)=-FK1 K2-1,$

The quantum group ${U}_{q}\left({SL}_{2}\right)$ is the subalgebra of ${U}_{q}\left({GL}_{2}\right)$ generated by $E,F$ and $K={K}_{1}{K}_{2}^{-1}\text{.}$ Alternatively, the quantum group ${U}_{q}\left({SL}_{2}\right)$ 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 ${U}_{q}\left({PGL}_{2}\right)$ is ${U}_{q}\left({GL}_{2}\right)$ with the additional relation ${K}_{1}{K}_{2}=1\text{.}$

In summary, there are Hopf algebra homomorphisms

$Uq(SL2) ↪ Uq(GL2) ⟶π Uq(PGL2) = Uq(GL2)⟨K1K2-1⟩ E⟼E⟼E F⟼F⟼F K⟼K1K2-1 K1K2⟼1$

The short exact sequences

$1⟶SL2⟶GL2 ⟶detℂ×⟶1and 1⟶ℂ×⟶GL2 ⟶πPGL2⟶1$

produce

$𝔥PGL* ↪ 𝔥GL* ⟶ 𝔥SL*= 𝔥GL*ℤ(ε1+ε2) kε1-kε2 ⟼ kε1-kε2 for k∈ℤ, aε1+bε2 ⟼ [(a-b)ε1]$

Thus

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

with

$ℂ[𝔥PGL*] ↪ ℂ[𝔥GL*] ⟶ ℂ[𝔥SL*]= ℂ[𝔥GL*] ⟨Y2=Y1-1⟩ Y ⟼ Y1Y2-1 Y1aY2b ⟼ Y1a-b$

The dominant weights

$(𝔥G*)+= { aε1+bε2 | a,b∈ℤ with a≥b }$

are the elements of ${𝔥}_{G}^{*}$ on the positive side of the hyperplane

$𝔥ε1∨-ε2∨= { aε1+bε2 | a,b∈ℤ with a-b=0 } .$

and the Weyl group ${W}_{0}=\left\{1,{s}_{1}\right\}$ has ${s}_{1}^{2}=1,$ with ${s}_{1}$ reflection in the hyperplane ${𝔥}^{{\epsilon }_{1}^{\vee }-{\epsilon }_{2}^{\vee }}\text{.}$ Then

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

and the representation theories of ${PGL}_{2},{GL}_{2}$ and ${SL}_{2}$ are related by

$π*:Rep (PGL2)↪ Rep(GL2)and Rep(GL2)= {deti | i∈ℤ} ⊗Rep(SL2)$

so that

$Rep(PGL2) ⟶π* Rep(GL2) ⟶res Rep(SL2)= Rep(GL2)⟨det⟩ L(kα) ⟼ L(k(ε1-ε2)) for k∈ℤ≥0, L(aε1+bε2) ⟼ L((a-b)ε1) for a≥b$

${\epsilon }_{2}^{\vee } {\epsilon }_{1}^{\vee }$