Symplectic linear map - Imported Knowlpedia Content

Definition

Let $(V,\omega_V)$ and $(W,\omega_W)$ be finite-dimensional symplectic vector spaces. A symplectic linear map is a linear map $T:V\to W$ satisfying

T^*\omega_W=\omega_V,

or equivalently

\omega_W(Tv,Tv')=\omega_V(v,v') \quad\text{for all }v,v'\in V.

Bijectivity is not part of the definition. Nevertheless, nondegeneracy of $\omega_V$ forces $T$ to be injective: if $Tv=0$ , then $\omega_V(v,v')=0$ for every $v'$ , hence $v=0$ . Thus $T$ identifies $V$ with a symplectic subspace of $W$ .

Functorial properties

Identity maps are symplectic linear, and composites of symplectic linear maps are symplectic linear because pullbacks compose. The image $T(V)$ carries the restricted form nondegenerately, and $T:V\to T(V)$ is a linear symplectomorphism. If $\dim V=\dim W$ , injectivity makes $T$ bijective.

Matrix criterion

Choose bases in which the two forms have matrices $J_V$ and $J_W$ , and let $A$ represent $T$ . The defining equation becomes

A^{\mathsf T}J_WA=J_V.

For the standard spaces $\mathbb R^{2m}\to\mathbb R^{2n}$ , this criterion shows directly that $m\leq n$ . When $m=n$ , its solutions are the matrices in the real symplectic group.

Examples and scope

The inclusion of a symplectic subspace with its restricted form is symplectic linear. A scalar multiple $cI$ on a nonzero real symplectic vector space is symplectic precisely when $c^2=1$ . This linear notion is the tangent-space model for a symplectic map between symplectic manifolds.

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.1, symplectic linear maps and matrices.
Maurice A. de Gosson, Symplectic Geometry and Quantum Mechanics, Birkhäuser, 2006. DOI record. Relevant: Chapter 1, symplectic spaces and the symplectic group.