Symplectic vector space

In linear algebra, a symplectic vector space or, for short, a symplectic space is a vector space together with a symplectic form , i.e. a non-degenerate alternating bilinear form . While the symmetrical bilinear form “ scalar product ” measures the length of vectors, the alternating bilinear form concerns the area size of the parallelogram spanned by two vectors .

definition

A symplectic vector space over a body is a vector space together with a bilinear form that has the following two properties: ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle \langle {-},{-}\rangle \colon V\times V\to K}$

• ${\displaystyle \langle {-},{-}\rangle }$is alternating , i.e. for everyone ${\displaystyle \langle v,v\rangle =0}$ ${\displaystyle v\in V}$
• ${\displaystyle \langle {-},{-}\rangle }$has not degenerated , i.e. for each there is a with ${\displaystyle 0\neq v\in V}$ ${\displaystyle w\in V}$ ${\displaystyle \langle v,w\rangle \neq 0\,.}$

A bilinear shape with these two properties is also called a symplectic shape. Because of

${\displaystyle 0=\langle v+w,v+w\rangle =\langle v,v\rangle +\langle v,w\rangle +\langle w,v\rangle +\langle w,w\rangle =\langle v,w\rangle +\langle w,v\rangle }$

the alternating form changes its sign when its arguments are exchanged .

Examples

• If a Hermitian scalar product is on a complex vector space V, then it is a symplectic form on V (understood as a real vector space). ${\displaystyle B({-},{-})}$ ${\displaystyle \langle v,w\rangle =\mathrm {Im} \,B(v,w)}$

• The hyperbolic planes form an important class of symplectic spaces : If V is two-dimensional with base {v, w}, and if V or the triple (V, v, w) is called a hyperbolic plane. It then applies ${\displaystyle \langle v,w\rangle =1}$

${\displaystyle \langle av+bw,cv+dw\rangle =ad-bc.}$

Classification of symplectic spaces

Every finite-dimensional symplectic vector space has dimension 2n and there is a basis with ${\displaystyle \{e_{1},\ldots ,e_{n},f_{1},\ldots ,f_{n}\}}$

${\displaystyle \langle e_{i},e_{j}\rangle =0}$

${\displaystyle \langle f_{i},f_{j}\rangle =0}$

${\displaystyle \langle e_{i},f_{j}\rangle =\delta _{ij}}$( Kronecker symbol ).

In particular, all symplectic spaces of dimension 2n are isometric . and set up a hyperbolic plane for every i, the entire symplectic space is therefore an orthogonal direct sum of hyperbolic planes. In physics, the elements e _i and f _{i are} called "canonically conjugated" (e.g. position or momentum variables) and the symplectic scalar product is identical to the so-called Poisson bracket . ${\displaystyle e_{i}}$ ${\displaystyle f_{i}}$

The automorphisms of a symplectic space form the symplectic group .

Symplectic manifold

→ Main article : Symplectic manifold

Symplectic vector spaces are the basis for the concept of symplectic manifold , which plays a role in the Hamilton formalism . Just like the symplectic vector spaces, the symplectic manifolds are also called symplectic spaces for short. Analogous to the symplectic bilinear form, there are also symplectic forms on these manifolds; these are special differential forms (a generalization of the alternating bilinear forms).

literature

• Rolf Berndt (mathematician) : Introduction to Symplectic Geometry (= Advanced Lectures in Mathematics ). Friedr. Vieweg & Sohn, Braunschweig / Wiesbaden 1998, ISBN 978-3-528-03102-2 .
• Rolf Berndt (mathematician) : An Introduction to Symplectic Geometry (= Graduate Studies in Mathematics 26). American Mathematical Society, Providence RI 2001, ISBN 0-8218-2056-7 .
• Serge Lang : Algebra (= Graduate Texts in Mathematics 211). Revised 3rd edition. Springer, New York NY et al. 2002, ISBN 0-387-95385-X , Chapter XV, § 8.

Individual evidence

1. ^ VI Arnold : Mathematical Methods of Classical Mechanics (= Graduate Texts in Mathematics 60). 2nd edition. Springer, New York NY et al. 1989, ISBN 0-387-96890-3 .