2012/02/14 by Yakimova, Oksana · 1 citation
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1202.3009
We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra A=K[W] 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 W (dxFi 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.