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,\ldots,F_r such that they give Kostant's regularity criterion on \mathbb A^n ( 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 \gt g corresponding to a decomposition \gt g=\gt h \oplus V , where \gt h is a subalgebra. Here {\mathcal A}={\mathcal S}(\gt g)=\mathbb K[\gt g^*] , Z({\mathcal A})={\mathcal S}(\gt g)^\gt g , and the contracted Lie algebra is a semidirect product of \gt h and an Abelian ideal isomorphic to \gt g/\gt h as an \gt h -module. In the first example, \gt h is a symmetric subalgebra and in the second, it is a Borel subalgebra and V is the nilpotent radical of an opposite Borel.