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symplectic vector space
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(Definition)
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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:
- Alternating: For all $v,w\in V$ $\omega(v,w)=-\omega(w,v)$
- 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. $$ \omega(Tv,Tw)=\omega(v,w)\ \ \forall v,w\in W.$$ 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].
- 1
- D. McDuff, D. Salamon, Introduction to Symplectic Topology, Clarendon Press, 1997.
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"symplectic vector space" is owned by matte. [ full author list (4) ]
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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
There are 3 references to this entry.
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 4724 times total.
Classification:
| AMS MSC: | 53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general) |
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Pending Errata and Addenda
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