2008/09/29 by Butin, Frédéric
#FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Rings and Algebras (math.RA) #Symbolic Computation (cs.SC)
paper · doi:10.48550/arxiv.0809.4983
Let \gog be a finite-dimensional semi-simple Lie algebra, \goh a Cartan subalgebra of \gog, and W its Weyl group. The group W acts diagonally on V:=\goh⊕\goh^*, as well as on ℂ[V]. The purpose of this article is to study the Poisson homology of the algebra of invariants ℂ[V]W endowed with the standard symplectic bracket. To begin with, we give general results about the Poisson homology space in degree 0, denoted by HP0(ℂ[V]W), in the case where \gog is of type Bn-Cn or Dn, results which support Alev's conjecture. Then we are focusing the interest on the particular cases of ranks 2 and 3, by computing the Poisson homology space in degree 0 in the cases where \gog is of type B2 (\goso5), D2 (\goso4), then B3 (\goso7), and D3=A3 (\goso6≃\gosl4). In order to do this, we make use of a functional equation introduced by Y. Berest, P. Etingof and V. Ginzburg. We recover, by a different method, the result established by J. Alev and L. Foissy, according to which the dimension of HP0(ℂ[V]W) equals 2 for B2. Then we calculate the dimension of this space and we show that it is equal to 1 for D2. We also calculate it for the rank 3 cases, we show that it is equal to 3 for B3-C3 and 1 for D3=A3.