We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra is said to be of Kostant type, if its centre is freely generated by homogeneous polynomials such that they give Kostant's regularity criterion on ( are linear independent if and only if the Poisson tensor has the maximal rank at ). If the initial Poisson algebra is of Kostant type and 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 corresponding to a decomposition , where is a subalgebra. Here , , and the contracted Lie algebra is a semidirect product of and an Abelian ideal isomorphic to as an -module. In the first example, is a symmetric subalgebra and in the second, it is a Borel subalgebra and is the nilpotent radical of an opposite Borel.
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Oksana Yakimova, One-parameter contractions of Lie-Poisson brackets. J. Eur. Math. Soc. 16 (2014), no. 2, pp. 387–407DOI 10.4171/JEMS/436