The group $\mathrm{Sp}\left(n\right)\simeq U_n\left(ℍ\right)$

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

Last updates: 5 November 2011

The group $\mathrm{Sp}\left(n\right)\simeq U_n\left(ℍ\right)$

The maximal compact subgroup of ${\mathrm{Sp}}_{2n}\left(ℂ\right)$ is

$\mathrm{Sp}\left(n\right)=U_{2n}\left(ℂ\right)\cap {\mathrm{Sp}}_{2n}\left(ℂ\right).$

The fundamental representation $\theta :U_1\left(ℍ\right)\stackrel{\sim }{⟶}{\mathrm{SU}}_2\left(ℂ\right).$

The action of ${ℍ}^{*}$ on the 2-dimensional $ℂ$-vector space $ℍ$ by right multiplication provides

$\begin{array}{cccccc}\theta :& {ℍ}^{*}& ⟶& {\mathrm{GL}}_2\left(ℂ\right)& \stackrel{\text{transpose}}{⟶}& {\mathrm{GL}}_2\left(ℂ\right)\\ & x_0+x_{1}i+x_{2}j+x_{3}k& ⟼& \left(\begin{array}{cc}{x}_0+x_{1}i& -x_2+x_{3}i\\ x_2+x_{3}i& x_0-x_{1}i\end{array}\right)& ⟼& \left(\begin{array}{cc}{x}_0+x_{1}i& -x_2+x_{3}i\\ x_2+x_{3}i& x_0-x_{1}i\end{array}\right)\end{array}$

This gives a group homomorphism

$\begin{array}{cccc}\theta :& {ℍ}^{*}& ⟶& {\mathrm{GL}}_2\left(ℂ\right)\\ & a+cj& ⟼& \left(\begin{array}{cc}a& c\\ -\bar{c}& \bar{a}\end{array}\right)\end{array}\text{for}a=x_0+x_{1}i\text{and}c=x_2+x_{3}i\text{in}ℂ.$

The Pauli matrices are

$\theta \left(i\right)=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),\theta \left(j\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),\theta \left(k\right)=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),$

and

$\theta :U_1\left(ℍ\right)\stackrel{\sim }{⟶}{\mathrm{SU}}_2\left(ℂ\right).$

More generally, for $n\in {\mathbb{Z}}_{>0}$, the function

$\begin{array}{cccc}\theta :& {\mathrm{GL}}_n\left(ℍ\right)& ⟶& {\mathrm{GL}}_{2n}\left(ℂ\right)\\ & \left(g_{ij}\right)& ⟼& \left(\theta \left(g_{ij}\right)\right)\end{array}\text{satisfies}\theta \left({\bar{g}}^t\right)={\overline{\theta \left(g\right)}}^t$

and is a group homomorphism. Restriction to $U_n\left(ℍ\right)$ gives an isomorphism,

$\theta :U_n\left(ℍ\right)\stackrel{\sim }{⟶}\mathrm{Sp}\left(n\right)=U_{2n}\left(ℂ\right)\cap {\mathrm{Sp}}_{2n}\left(ℂ\right)$

where

$U_n\left(ℍ\right)=\{g\in {\mathrm{GL}}_n(ℍ\left)\right|g{\bar{g}}^t=1\},U_{2n}\left(ℂ\right)=\{g\in {\mathrm{GL}}_{2n}(ℂ\left)\right|g{\bar{g}}^t=1\}$

and

${\mathrm{Sp}}_{2n}\left(ℂ\right)=\{g\in {\mathrm{GL}}_{2n}(ℂ\left)\right|gJ{g}^t=J\},\text{where}J=\left(\begin{array}{cc}0& \begin{array}{ccc}1& & 0\\ & \ddots & \\ 0& & 1\end{array}\\ \begin{array}{ccc}-1& & 0\\ & \ddots & \\ 0& & -1\end{array}& 0\end{array}\right)$

${\mathrm{Sp}}_2\left(ℂ\right)={\mathrm{SL}}_2\left(ℂ\right)$.

Notes and References

These notes were influenced by the Wikipedia articles ????. They were prepared for lectures and working seminars in Representation Theory at University of Melbourne in 2008-2011.

References

[St] R. Steinberg, Lectures on Chevalley groups, Notes prepared by John Faulkner and Robert Wilson, Yale University, New Haven, Conn., 1968. iii+277 pp. MR0466335.

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