## The group ${\mathrm{Sp}}_{4}$ and the Lie algebra ${\mathrm{𝔰𝔭}}_{4}$

Last update: 12 May 2012

## The group ${\mathrm{Sp}}_{4}$

The symplectic group ${\mathrm{Sp}}_{4}\left(ℂ\right)$ is $Sp4(ℂ) = {g∈ GL4(ℂ) | gtJg=J }, where J= ( 0 I2 -I2 0 ) and I2 = ( 1 0 0 1 ) .$ Let $𝔤={\mathrm{𝔰𝔭}}_{4}\left(ℂ\right)$ be the Kac-Moody algebra with generalised Cartan matrix $A=\left(\begin{array}{cc}2& -2\\ -1& 2\end{array}\right)$. Then $𝔤= 𝔰𝔭4(ℂ) ={ x∈Mat4(ℂ) | Jx+xtJ =0 }$ is the span of the elements $Xα1 =E12-E43 , Xα2 =E24, Xα1 +α2 = E14+E23, X2α1+α2 =E13, X-α1 =E21-E34 , X-α2 =E42, X-α1-α2 =E41+E32 , X-2α1-α2 =E31,$ $Hα1∨ = E11-E22 -E33+E44 , and Hα2∨ =E22-E44 ,$ where ${E}_{ij}$ is the $4×4$ matrix with a 1 in the $ij-$entry and zeros elsewhere. Here the root system $R$ is of type${C}_{2},$ with $R+ ={ α1= e1-e2, α2= 2e2, α1+α2, 2α1+α2 }.$ Let $V$ be the 4-dimensional representation of ${\mathrm{𝔰𝔭}}_{4}\left(ℂ\right)$ on column vectors. Therefore $xα1(f) = ( 1f00 0100 0010 00-f1 ) , xα2(f) = ( 1000 010f 0010 0001 ) ,$ $xα1 +α2 (f) = ( 100f 01f0 0010 0001 ) , and x2 α1+α2 (f) = ( 10f0 0100 0010 0001 ) .$ We compute $nα1(f) = ( 0f00 -f-1 000 000 f-10 0-f0 ), and nα2(f) = ( 1000 000f 0010 0 -f-10 0 ),$ and therefore $hα1∨ (f) = ( f000 0f-1 0000 f-10 000f ), and hα2∨ (f) = ( 1000 0f00 0010 000 f-1 ),$ and $G={\mathrm{Sp}}_{4}$. The nontrivial commutator relations are $xα1 (f1) xα2 (f2) = xα2 (f2) xα1 (f1) xα1+α2 (f1 f2) x2α1 +α2 (-f12 f2) xα1 (f1)x α1 +α2 (f2) = xα1 +α2 (f2) xα1 (f1) x2α1 +α2 (2f1 f2).$

## Notes and References

These notes are adapted from the lecture notes of Arun Ram on representation theory, from 2008.

References?