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symplectic vector space (Definition)

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. 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 symplectic form for $ V$.

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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symplectic complement (Definition) by matte
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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:
AMS MSC53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general)

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