nLab affine symplectic group



Group Theory

Symplectic geometry



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.



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) }


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.