# 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 ${\mathcal A}=\mathbb K[\mathbb A^n]$ 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.

## 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