Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities (Q2898923)

This paper concentrates on surfaces \(X\) with an isolated singularity at the origin and on the n-th symmetric power \(Y\) of \(X\). Let us recall that such surfaces were studied by Saito in 1987. In this work, the authors compute explicitly the zeroth Poisson homology of the algebra of functions on \(Y=X^n\), as a graded vector space using the weight grading. In particular, the main result of the paper confirms a conjecture of Alev about the Kleinian case. Moreover, in the elliptic case, the authors are able to describe the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators. This can be useful to study the quantizations of \(Y\).