Decomposition of (co)isotropic relations

TL;DR

This paper classifies indecomposable coisotropic relations between Poisson vector spaces, showing they can be decomposed into basic building blocks and identifying invariants that classify these decompositions.

Contribution

It provides a complete classification of indecomposable coisotropic relations and establishes a duality-based approach for Poisson vector spaces.

Findings

Thirteen isomorphism classes of indecomposable coisotropic relations identified.

Every coisotropic relation decomposes into a direct sum of these indecomposables.

Thirteen invariants form a basis for classifying these relations.

Abstract

We identify thirteen isomorphism classes of indecomposable coisotropic relations between Poisson vector spaces and show that every coisotropic relation between finite-dimensional Poisson vector spaces may be decomposed as a direct sum of multiples of these indecomposables. We also find a list of thirteen invariants, each of which is the dimension of a space constructed from the relation, such that the 13-vector of multiplicities and the 13-vector of invariants are related by an invertible matrix over $\mathbb Z$. It turns out to be simpler to do the analysis above for isotropic relations between presymplectic vector spaces. The coisotropic/Poisson case then follows by a simple duality argument.