nLab
affine symplectic group

Contents

Context

Group Theory

Symplectic geometry

Contents

Definition

Given a symplectic vector space (V,ω)(V,\omega), then its affine symplectic group ASp(V,ω)ASp(V,\omega) (or inhomogeneous sympelctic group ISp(V,ω)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,ω)ESp(V,\omega) of the affine symplectic group – the extended affine symplectic group – given by restricting the quantomorphism group of (V,ω)(V,\omega) to affine transformations. The further restriction of that to elements coming from translations is the Heisenberg group Heis(V,ω)Heis(V,\omega).

Heis(V,ω) ESp(V,ω) QuantMorph(V,ω) V ASp(V,ω) HamSympl(V,ω) \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.