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BIRS Workshop Lecture Videos

Hopf Actions on AS Regular Algebras: Auslander's Theorem Kirkman, Ellen


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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