One-parameter contractions of Lie-Poisson brackets

Abstract

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $\mathcal{A}=\mathbb{K}[{\mathbb{A}}^n]$ is said to be of Kostant type, if its centre $Z(\mathcal{A})$ is freely generated by homogeneous polynomials $F_1,\dots ,F_r$ such that they give Kostant's regularity criterion on ${\mathbb{A}}^n$ ($d_x{F}_i$ 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 $\mathcal{A}=\mathcal{S}(>g)=\mathbb{K}[>g^{\ast }]$, $Z(\mathcal{A})=\mathcal{S}(>g{)}^{>}g$, 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.