# Koszul properties of the moment map of some classical representations

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## 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.

## Keywords

Betti number Classical Lie algebra Koszul algebra Moment map Poincaré series Standard representation## Mathematics Subject Classification

13D02 16S37 53D20## Notes

### Acknowledgements

Our thanks to Lucho Avramov for helpful conversations regarding this work; in particular, for pointing out Lemma 3.3, and the work of Hreinsdottir [13]. Part of this article is based on work supported by the National Science Foundation under Grant No. 0932078000, while AC and SBI were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the 2012–2013 Special Year in Commutative Algebra. AC was supported by INdAM-GNSAGA and PRIN “Geometry of Algebraic Varieties” 2015EYPTSB_008. SBI was partly supported by NSF Grants DMS-1503044.

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