One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova

Journal of the European Mathematical Society, Tome 16 (2014) no. 2, pp. 387-407.

Résumé

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra A=K[An] is said to be of Kostant type, if its centre Z(A) is freely generated by homogeneous polynomials F1​,...,Fr​ such that they give Kostant's regularity criterion on An (dx​Fi​ 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 Fi​ 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⊕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.

Détail

DOI : 10.4171/jems/436

Classification : 17-XX
Keywords: Nilpotent orbits, centralisers, symmetric invariants