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Hopf Actions on AS Regular Algebras: Auslander's Theorem Kirkman, Ellen
Description
Let ${\Bbbk}$ be an algebraically closed field of characteristic zero. Maurice Auslander proved that when a finite subgroup $G$ of GL$_n(\Bbbk)$, containing no reflections, acts on $A=\Bbbk[x_1,\dots,x_n]$ naturally, with fixed subring $A^G$, then the skew group algebra $A \# G$ is isomorphic to End$_{A^G}(A)$ as algebras. There are recent results extending Auslander's Theorem to non(co)commutative settings of actions on Artin-Schelter regular algebras $A$ by groups or Hopf algebras that contain no ``reflections''. Bao, He, and Zhang develop the notion of pertinency, and apply it to prove Auslander's Theorem for certain group actions on $\Bbbk_{-1}[x_1, \dots, x_n]$, on $U(\mathfrak{g})$ for $\mathfrak{g}$ finite dimensional, and on certain classes of noetherian down-up algebras. Work in progress with Gaddis and Moore proves Auslander's Theorem for the permutation action of $S_n$ on $\Bbbk_{-1}[x_1, \dots, x_n]$ for n= 3 and 4. Work with Chan, Walton and Zhang proves Auslander's theorem when $A$ is an AS regular algebra of dimension 2 and $H$ is a semisimple Hopf algebra acting on $A$ so that $A$ is a graded $H$-module algebra under an action that is inner faithful and has trivial homological determinant. With Chen and Zhang we prove Auslander's Theorem for homogeneous, inner-faithful group coactions on noetherian down-up algebras with trivial homological determinant.
Item Metadata
Title |
Hopf Actions on AS Regular Algebras: Auslander's Theorem
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2016-09-13T09:01
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Description |
Let ${\Bbbk}$ be an algebraically closed field of characteristic zero. Maurice Auslander proved that when a finite subgroup $G$ of GL$_n(\Bbbk)$, containing no reflections, acts on $A=\Bbbk[x_1,\dots,x_n]$ naturally, with fixed subring $A^G$, then the skew group algebra $A \# G$ is isomorphic to End$_{A^G}(A)$ as algebras. There are recent results extending Auslander's Theorem to non(co)commutative settings of actions on Artin-Schelter regular algebras $A$ by groups or Hopf algebras that contain no ``reflections''. Bao, He, and Zhang develop the notion of pertinency, and apply it to prove Auslander's Theorem for certain group actions on
$\Bbbk_{-1}[x_1, \dots, x_n]$, on $U(\mathfrak{g})$ for $\mathfrak{g}$ finite dimensional, and on certain classes of noetherian down-up algebras. Work in progress with Gaddis and Moore proves Auslander's Theorem for the permutation action of $S_n$ on
$\Bbbk_{-1}[x_1, \dots, x_n]$ for n= 3 and 4.
Work with Chan, Walton and Zhang proves Auslander's theorem when $A$ is an AS regular algebra of dimension 2 and $H$ is a semisimple Hopf algebra acting on $A$ so that $A$ is a graded $H$-module algebra under an action that is inner faithful and has trivial homological determinant. With Chen and Zhang we prove Auslander's Theorem for homogeneous, inner-faithful group coactions on noetherian down-up algebras with trivial homological determinant.
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Extent |
55 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Wake Forest University
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Series | |
Date Available |
2017-03-15
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0343183
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International