2019
DOI: 10.1090/proc/14363
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Auslander’s Theorem for permutation actions on noncommutative algebras

Abstract: When A = k[x 1 , . . . , xn] and G is a small subgroup of GLn(k), Auslander's Theorem says that the skew group algebra A#G is isomorphic to 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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Cited by 23 publications
(21 citation statements)
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“…As a consequence of [GKMW,Theorem 3.4], if R G is a graded isolated singularity, then so is R H for all subgroups 1 H ⊆ G. The graded isolated singularities in the above lemma are all conventional. One nice example of non-conventional graded isolated singularities is given by Gaddis-Kirkman-Moore-Won [GKMW].…”
Section: More Examples Of Graded Isolated Singularitiesmentioning
confidence: 91%
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“…As a consequence of [GKMW,Theorem 3.4], if R G is a graded isolated singularity, then so is R H for all subgroups 1 H ⊆ G. The graded isolated singularities in the above lemma are all conventional. One nice example of non-conventional graded isolated singularities is given by Gaddis-Kirkman-Moore-Won [GKMW].…”
Section: More Examples Of Graded Isolated Singularitiesmentioning
confidence: 91%
“…This implies that σ fixes one x i . Replacing G by the subgroup σ , we can assume that G = σ following [GKMW,Theorem 3.4]. Since σ fixes x i , one can show that x N i is not in r(R, G) for all N ≥ 0 (which also follows from Lemma 10.4(6) in an appropriate setting).…”
Section: More Examples Of Graded Isolated Singularitiesmentioning
confidence: 99%
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