Auslander's Theorem for permutation actions on noncommutative algebras

When \(A = \mathbb{k}[x_1, \ldots, x_n]\) and \(G\) is a small subgroup of \(\operatorname{GL}_n(\mathbb{k})\), 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 Theo...

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Bibliographic Details
Published in:arXiv.org 2018-09
Main Authors: Gaddis, Jason, Kirkman, Ellen, Moore, W Frank, Won, Robert
Format: Article
Language:English
Subjects:
Algebra
Permutations
Polynomials
Rings (mathematics)
Singularities
Subgroups
Theorems
Online Access:Get full text
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Summary:When \(A = \mathbb{k}[x_1, \ldots, x_n]\) and \(G\) is a small subgroup of \(\operatorname{GL}_n(\mathbb{k})\), 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 graded down-up algebra. We also show that certain fixed rings \(A^G\) are graded isolated singularities in the sense of Ueyama.
ISSN:2331-8422
DOI:10.48550/arxiv.1705.00068