Auslander’s Theorem for permutation actions on noncommutative algebras

Gaddis, Jason; Kirkman, Ellen; Moore, W.; Won, Robert

doi:10.1090/proc/14363

Auslander’s Theorem for permutation actions on noncommutative algebras
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by Jason Gaddis, Ellen Kirkman, W. Frank Moore and Robert Won PDF

Proc. Amer. Math. Soc. 147 (2019), 1881-1896 Request permission

Abstract:

When $A = \Bbbk [x_1, \ldots , x_n]$ and $G$ is a small subgroup of $\operatorname {GL}_n(\Bbbk )$, Auslander’s Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname {End}_{A^G}(A)$ 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 homogeneous 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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