symplectic vector space

A symplectic vector space $(V,\omega )$ is a finite dimensional real vector space $V$ equipped with an alternating non-degenerate 2-tensor, i.e., a bilinear map $\omega :V\times V\to ℝ$ that satisfies the following properties:

1. Alternating: For all $v,w\in V$, $\omega (v,w)=-\omega (w,v)$.

2. Non-degenerate: If $\omega (v,w)=0$ for all $w\in V$, then $v=0$.

The tensor $\omega$ is called a for $V$.

A linear automorphism $T\in \mathrm{Aut}(V)$ is called linear symplectomorphism when $T^{*}\omega =\omega$, i.e.

$$\omega (Tv,Tw)=\omega (v,w)\mathit{\hspace{1em}}\forall v,w\in W.$$

Linear symplectomorphisms of $(V,\omega )$ form a group (under composition of linear map) that is denoted by $\mathrm{Sp}(\mathrm{V},\omega )$.

One can show that a symplectic vector space is always even dimensional [1].

Title	symplectic vector space
Canonical name	SymplecticVectorSpace
Date of creation	2013-03-22 13:32:22
Last modified on	2013-03-22 13:32:22
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	11
Author	matte (1858)
Entry type	Definition
Classification	msc 53D05
Defines	symplectic vector space
Defines	linear symplectomorphism