nLab affine symplectic group

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 intersection $ASp(V,\omega) = Aff(V)\times_{Diff(V)} Sympl(X,\omega)$ of the affine group of the affine space $V$ and the symplectomorphism group of the symplectic manifold $(X,\omega)$, i.e.the group of all those affine transformations which preserve the symplectic form $\omega$;

• the semidirect product $ASp(V,\omega) = V \rtimes Sp(V,\omega)$ of the symplectic group acting on $V$ regarded as the translation group over itself.

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 $Heis(V,\omega)$.

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

Related concepts

• Poincare group

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)