# 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.

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