# 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 $\omega\colon V\times V\rightarrow\mathbb{R}$ that satisfies the following properties:

1. 1.

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

2. 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.

 $\omega(Tv,Tw)=\omega(v,w)\ \ \forall v,w\in W.$

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

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

## References

• 1 D. McDuff, D. Salamon, Introduction to Symplectic Topology, Clarendon Press, 1997.
Title symplectic vector space SymplecticVectorSpace 2013-03-22 13:32:22 2013-03-22 13:32:22 matte (1858) matte (1858) 11 matte (1858) Definition msc 53D05 symplectic vector space linear symplectomorphism