# Moscow Mathematical Journal

Volume 21, Issue 2, April–June 2021  pp. 383–399.

Goldie Ranks of Primitive Ideals and Indexes of Equivariant Azumaya Algebras

Authors:  Ivan Losev (1) and Ivan Panin (2)
Author institution:(1) Department of Mathematics, Yale University, New Haven, CT, USA
(2) St. Petersburg branch of V.A. Steklov Mathematical Institute, St. Petersburg, Russian Federation

Summary:

Let $\mathfrak{g}$ be a semisimple Lie algebra. We establish a new relation between the Goldie rank of a primitive ideal $\mathcal{J}\subset U(\mathfrak{g})$ and the dimension of the corresponding irreducible representation $V$ of an appropriate finite W-algebra. Namely, we show that $\operatorname{Grk}(\mathcal{J}) \leqslant \dim V/d_V$, where $d_V$ is the index of a suitable equivariant Azumaya algebra on a homogeneous space. We also compute $d_V$ in representation theoretic terms.

2020 Math. Subj. Class. 17B35, 16H99

Keywords: Azumaya algebras, index, primitive ideals, Goldie ranks, W-algebras.