# nLab affine symplectic group

Contents

group theory

### Cohomology and Extensions

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Definition

Given a symplectic vector space $(V,\omega)$, then its affine symplectic group $ASp(V,\omega)$ (or inhomogeneous sympelctic group $ISp(V,\omega)$) is equivalently

The further restriction to linear functions gives the symplectic group proper.

## Properties

### Extensions

There is a circle group extension $ESp(V,\omega)$ of the affine symplectic group – the extended affine symplectic group – given by restricting the quantomorphism group of $(V,\omega)$ to affine transformations. The further restriction of that to elements coming from translations is the Heisenberg group $Heis(V,\omega)$.

$\array{ Heis(V,\omega) &\hookrightarrow& ESp(V,\omega) &\hookrightarrow& QuantMorph(V,\omega) \\ \downarrow && \downarrow && \downarrow \\ V &\hookrightarrow& ASp(V,\omega) &\hookrightarrow& HamSympl(V,\omega) }$

## References

Review includes

• Stephen G. Low, section 1 of Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group, J. Math. Phys. 55, 022105 (2014) (arXiv:1207.6787)

Last revised on January 2, 2015 at 14:22:03. See the history of this page for a list of all contributions to it.