The group ${\mathrm{Sp}}_4$ and the Lie algebra ${\mathrm{𝔰𝔭}}_4$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 12 May 2012

The group ${\mathrm{Sp}}_4$

The symplectic group ${\mathrm{Sp}}_4\left(ℂ\right)$ is $${\mathrm{Sp}}_4\left(ℂ\right)=\{g\in {\mathrm{GL}}_4(ℂ\left)\right|g^{t}Jg=J\},\text{where}J=\left(\begin{array}{cc}0& I_2\\ -I_2& 0\end{array}\right)\text{and}{I}_2=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right).$$ 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 $$𝔤={\mathrm{𝔰𝔭}}_4\left(ℂ\right)=\{x\in {\mathrm{Mat}}_4(ℂ\left)\right|Jx+x^{t}J=0\}$$ is the span of the elements $$\begin{array}{rrrr}{X}_{{\alpha }_1}=E_{12}-E_{43},& X_{{\alpha }_2}=E_{24},& X_{{\alpha }_1+{\alpha }_2}=E_{14}+E_{23},& X_{2{\alpha }_1+{\alpha }_2}=E_{13},\\ X_{-{\alpha }_1}=E_{21}-E_{34},& X_{-{\alpha }_2}=E_{42},& X_{-{\alpha }_1-{\alpha }_2}=E_{41}+E_{32},& X_{-2{\alpha }_1-{\alpha }_2}=E_{31},\end{array}$$ $$H_{{\alpha }_1^{\vee }}=E_{11}-E_{22}-E_{33}+E_{44},\text{and}{H}_{{\alpha }_2^{\vee }}=E_{22}-E_{44},$$ where $E_{ij}$ is the $4\times 4$ matrix with a 1 in the $ij-$entry and zeros elsewhere. Here the root system $R$ is of type$C_2,$ with $$R^{+}=\{{\alpha }_1=e_1-e_2,{\alpha }_2=2e_2,{\alpha }_1+{\alpha }_2,2{\alpha }_1+{\alpha }_2\}.$$ Let $V$ be the 4-dimensional representation of ${\mathrm{𝔰𝔭}}_4\left(ℂ\right)$ on column vectors. Therefore $$x_{{\alpha }_1}\left(f\right)=\left(\begin{array}{cccc}1& f& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& -f& 1\end{array}\right),x_{{\alpha }_2}\left(f\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& f\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),$$ $$x_{{\alpha }_1+{\alpha }_2}\left(f\right)=\left(\begin{array}{cccc}1& 0& 0& f\\ 0& 1& f& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right),\text{and}{x}_{2{\alpha }_1+{\alpha }_2}\left(f\right)=\left(\begin{array}{cccc}1& 0& f& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).$$ We compute $$n_{{\alpha }_1}\left(f\right)=\left(\begin{array}{cccc}0& f& 0& 0\\ -f^{-1}& 0& 0& 0\\ 0& 0& 0& f^{-1}\\ 0& 0& -f& 0\end{array}\right),\text{and}{n}_{{\alpha }_2}\left(f\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& f\\ 0& 0& 1& 0\\ 0& -f^{-1}& 0& 0\end{array}\right),$$ and therefore $$h_{{\alpha }_1^{\vee }}\left(f\right)=\left(\begin{array}{cccc}f& 0& 0& 0\\ 0& f^{-1}& 0& 0\\ 0& 0& f^{-1}& 0\\ 0& 0& 0& f\end{array}\right),\text{and}{h}_{{\alpha }_2^{\vee }}\left(f\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& f& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& f^{-1}\end{array}\right),$$ and $G={\mathrm{Sp}}_4$. The nontrivial commutator relations are $$\begin{array}{rcl}{x}_{{\alpha }_1}\left(f_1\right)x_{{\alpha }_2}\left(f_2\right)& =& x_{{\alpha }_2}\left(f_2\right)x_{{\alpha }_1}\left(f_1\right)x_{{\alpha }_1+{\alpha }_2}\left(f_1{f}_2\right)x_{2{\alpha }_1+{\alpha }_2}\left(-f_1^2{f}_2\right)\\ x_{{\alpha }_1}\left(f_1\right)x_{{\alpha }_1+{\alpha }_2}\left(f_2\right)& =& x_{{\alpha }_1+{\alpha }_2}\left(f_2\right)x_{{\alpha }_1}\left(f_1\right)x_{2{\alpha }_1+{\alpha }_2}\left(2f_1{f}_2\right).\end{array}$$

Notes and References

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

References

References?

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