# One-parameter contractions of Lie-Poisson brackets

### Oksana Yakimova

Friedrich-Schiller-Universität Jena, Germany

## Abstract

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $A=K[A_{n}]$ 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 $A_{n}$ ($d_{x}F_{i}$ 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 $>g=>h⊕V$, 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.

## Cite this article

Oksana Yakimova, One-parameter contractions of Lie-Poisson brackets. J. Eur. Math. Soc. 16 (2014), no. 2, pp. 387–407

DOI 10.4171/JEMS/436