One-parameter contractions of Lie-Poisson brackets

TL;DR

This paper studies how Lie-Poisson brackets behave under contractions, focusing on the preservation of algebraic structures and centers, with examples involving contractions of simple Lie algebras related to subalgebra decompositions.

Contribution

It establishes conditions under which Kostant-type polynomial Poisson algebras remain of Kostant type after contraction, with illustrative examples involving Lie algebra decompositions.

Findings

Centers of contracted algebras are preserved under certain conditions.

Kostant's regularity criterion is maintained in specific contractions.

Examples include contractions of simple Lie algebras with subalgebra decompositions.

Abstract

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra A=K[W] is said to be of Kostant type, if its centre Z(A) is freely generated by homogeneous polynomials F_1,...,F_r such that they give Kostant's regularity criterion on W (d_xF_i are linear independent if and only if the Poisson tensor has the maximal rank at x). If the initial Poisson algebra is of Kostant type and F_i satisfy a certain degree-equality, then the contraction is also of Kostant type. The general result is illustrated by two examples. Both are contractions of a simple Lie algebra g corresponding to a decomposition g=h \oplus V, where h is a subalgebra. Here A=S(g)=K[g^*], Z(A)=S(g)^g, and the contracted Lie algebra is a semidirect product of h and an Abelian ideal isomorphic to g/h as an h-module. In the first example, h is a symmetric subalgebra and in the second, it is a Borel subalgebra and V is the nilpotent radical of an opposite Borel.