Symplectic vector space - Imported Knowlpedia Content

Definition

A symplectic vector space is a pair $(V,\omega)$ consisting of a finite-dimensional real vector space $V$ and a bilinear form $\omega:V\times V\to\mathbb R$ such that:

$\omega(v,v)=0$ for every $v\in V$ , so $\omega$ is alternating; and
if $\omega(v,w)=0$ for every $w\in V$ , then $v=0$ , so $\omega$ is nondegenerate.

Equivalently, the linear map $V\to V^*$ , $v\mapsto\omega(v,\mathord{-})$ , is an isomorphism. No inner product, complex structure, or preferred basis is part of the data.

Normal form and dimension

Every symplectic vector space has even dimension $2n$ and admits a basis $e_1,\ldots,e_n,f_1,\ldots,f_n$ with

\omega(e_i,e_j)=\omega(f_i,f_j)=0,\qquad \omega(e_i,f_j)=\delta_{ij}.

Consequently, all real symplectic vector spaces of a fixed finite dimension are symplectically isomorphic. This standard normal form is proved in Cannas da Silva, §1.1.

Subspaces

For a subspace $W\subseteq V$ , its symplectic orthogonal is

W^\omega=\{v\in V:\omega(v,w)=0\text{ for every }w\in W\}.

The relations $W\subseteq W^\omega$ , $W^\omega\subseteq W$ , and $W=W^\omega$ define isotropic, coisotropic, and Lagrangian subspaces, respectively. A Lagrangian subspace has dimension $n$ .

Conventions and scope

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.1, symplectic vector spaces, normal forms, and subspaces.
Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. Publisher record. Relevant: Chapter 1, linear symplectic geometry.