Koszul properties of the moment map of some classical representations

Abstract

This work concerns the moment map $μ$ associated with the standard representation of a classical Lie algebra. For applications to deformation quantization it is desirable that $S/(μ)$, the coordinate algebra of the zero fibre of $μ$, be Koszul. The main result is that this algebra is not Koszul for the standard representation of $\mathfrak{sl}_{n}$, and of $\mathfrak{sp}_{n}$. This is deduced from a computation of the Betti numbers of $S/(μ)$ as an $S$-module, which are of interest also from the point of view of commutative algebra.

Revised version. Differences to version 1: title slightly changed, comments added at the end, minor revisions