Poisson manifold

As a Poisson manifold is known in the mathematics a differentiable manifold that with a Poisson structure is provided. A Poisson structure is a bilinear mapping on the algebra of smooth functions that fulfills the properties of a Poisson bracket . The Poisson manifold, structure and bracket are named after the physicist and mathematician Siméon Denis Poisson .

definition

A Poisson structure on a differentiable manifold is a bilinear map ${\displaystyle M}$

${\displaystyle \{\cdot ,\cdot \}_{M}\colon C^{\infty }(M)\times C^{\infty }(M)\to C^{\infty }(M)}$,

so that the bracket is antisymmetric

${\displaystyle \{f,g\}=-\{g,f\}}$,

is, the Jacobi identity

${\displaystyle \{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0}$

suffices and represents a derivation for everyone ${\displaystyle f,g,h\in C^{\infty }(M)}$

${\displaystyle \{fg,h\}=f\{g,h\}+\{f,h\}g}$.

The bilinear mapping of the Poisson structure is called a Poisson bracket and a differentiable manifold with a Poisson structure is called a Poisson manifold. ${\displaystyle \{\cdot ,\cdot \}}$

example

Let be a Lie algebra with Lie bracket and its dual space with the pairing . On can for through ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle {\mathfrak {g}}^{*}}$ ${\displaystyle \langle \cdot ,\cdot \rangle \colon {\mathfrak {g}}^{*}\times {\mathfrak {g}}\to \mathbb {R} }$ ${\displaystyle {\mathfrak {g}}^{*}}$ ${\displaystyle F,G\colon {\mathfrak {g}}^{*}\to \mathbb {R} }$

${\displaystyle \{F,G\}_{\pm }(\mu ):=\pm \left\langle \mu ,\left[{\frac {\delta F}{\delta \mu }},{\frac {\delta G}{\delta \mu }}\right]\right\rangle }$

can be explained with Poisson brackets. With the here is functional derivative of after referred to. The bracket is called a Lie-Poisson bracket. Together with this Poisson bracket, it becomes a Poisson manifold. This statement is called the Lie-Poisson theorem. ${\displaystyle \mu \in {\mathfrak {g}}^{*}}$ ${\displaystyle {\frac {\delta F}{\delta \mu }}\in {\mathfrak {g}}}$ ${\displaystyle F}$ ${\displaystyle \mu }$ ${\displaystyle \{\cdot ,\cdot \}_{\pm }}$ ${\displaystyle {\mathfrak {g}}^{*}}$

Applications

In particular, every symplectic manifold is also a Poisson manifold. In this case then is the defining structure

${\displaystyle \{f,g\}:=\sum _{ij}\omega ^{ij}\,\partial _{i}f\,\partial _{j}g}$

given by a 2-form or its components in local coordinates. ${\displaystyle \textstyle \omega =\sum _{ij}\omega _{ij}\,\mathrm {d} x^{i}\,\mathrm {d} x^{j}}$ ${\displaystyle \omega _{ij}}$

Poisson manifolds can be viewed as an algebraic abstraction from symplectic manifolds. In addition to a much larger class of morphisms, there are also differences, for example, in the fact that the condition that the Poisson bracket should not be singular anywhere, i.e. have full rank, is dropped.

This calculation is used, for example, in deformation theory. There it offers access to non-commutative geometry and geometric quantization.

Individual evidence

1. ^ R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 , pp. 609-610.
2. ^ R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 , p. 613.