symplectic vector space

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

  1. 1.

    Alternating: For all v,wV, ω(v,w)=-ω(w,v).

  2. 2.

    Non-degenerate: If ω(v,w)=0 for all wV, then v=0.

The tensor ω is called a for V.

A linear automorphismPlanetmathPlanetmath TAut(V) is called linear symplectomorphism when T*ω=ω, i.e.


Linear symplectomorphisms of (V,ω) form a group (under compositionMathworldPlanetmathPlanetmath of linear map) that is denoted by Sp(V,ω).

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


  • 1 D. McDuff, D. Salamon, Introduction to Symplectic Topology, Clarendon Press, 1997.
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