Auslander's Theorem for permutation actions on noncommutative algebras

Abstract

When $A=\mathbb{k}[x_1,\dots ,x_n]$ and $G$ is a small subgroup of ${\mathrm{GL}}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A\#G$ is isomorphic to ${\mathrm{End}}_{A^G}(A)$ as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on $(-1)$-skew polynomial rings, $(-1)$-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain graded down-up algebra. We also show that certain fixed rings $A^G$ are graded isolated singularities in the sense of Ueyama.

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