Symplectic vector space

A symplectic vector space or short symplectic space in linear algebra is a vector space together with a symplectic form, ie a non-degenerate alternating bilinear form. While the symmetric bilinear " scalar " measures the length of vectors alternating bilinear relates to the area size of the two vectors spanned parallelogram.

Definition

A symplectic vector space over a field is a vector space together with a bilinear form which has the following two properties:

• Is alternating, that is, for all
• Is non-degenerate, that is, for each, there exists a with

A bilinear form with these two properties is also called the symplectic form. because of

Changes the alternating form with permutation of their arguments sign.

Examples

• If a Hermitian inner product on a complex vector space V, is a symplectic form on V ( considered as a real vector space ).

• An important class of symplectic spaces form the hyperbolic planes: If V is two-dimensional with basis { v, w}, and is so called V or the triple ( V, v, w) is a hyperbolic plane. It is then

Classification of symplectic spaces

Every finite symplectic vector space has dimension 2n straight, and there is a basis with

In particular, all symplectic space of dimension 2n are isometric. and tension for each i to a hyperbolic plane, the whole symplectic space is therefore an orthogonal direct sum of hyperbolic planes. In physics, the elements are ei and fi referred to as "canonical - conjugated " (for example, local variables or pulse ) and the symplectic dot is identical with the so-called Poisson bracket.

The automorphisms of a symplectic space form the symplectic group.

Symplectic manifold

Symplectic vector spaces are the basis for the concept of symplectic manifold, which plays a role in the Hamiltonian formalism. Just as the symplectic vector spaces and the symplectic manifolds symplectic spaces are briefly mentioned. In analogy to the symplectic bilinear form is available on these manifolds also symplectic forms, these are special differential forms, which are a generalization of the alternating bilinear forms.