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 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.