# Koszul properties of the moment map of some classical representations

## TL;DR

This paper investigates the Koszul property of the coordinate algebra of the zero fiber of the moment map for classical Lie algebra representations, showing it fails for certain cases relevant to deformation quantization.

## Contribution

It provides the first explicit computations of Betti numbers for these algebras, demonstrating their non-Koszul nature in key classical cases.

## Key findings

- The algebra is not Koszul for the standard representation of sl(n) and sp(n).
- Betti number computations reveal the algebra's homological properties.
- Results impact understanding of deformation quantization and algebraic structures of moment maps.

## Abstract

This work concerns the moment map $\mu$ associated with the standard representation of a classical Lie algebra. For applications to deformation quantization it is desirable that $S/(\mu)$, the coordinate algebra of the zero fibre of $\mu$, 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 $S/(\mu)$ as an $S$-module, which are of interest also from the point of view of commutative algebra.