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symplectic vector space

(Definition)

A symplectic vector space $(V,\omega)$ is a finite dimensional real vector space 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:

The tensor $\omega$ is called a symplectic form for

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

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

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].

Bibliography

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

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Also defines:

symplectic vector space, linear symplectomorphism

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Cross-references: even, linear map, composition, group, automorphism, tensor, properties, bilinear map, non-degenerate, alternating, vector space, real, finite dimensional
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This is version 8 of symplectic vector space, born on 2003-04-02, modified 2006-10-15.
Object id is 4138, canonical name is SymplecticVectorSpace.
Accessed 3497 times total.

Classification:

53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)

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