Abstract
We consider the action of a semisimple Hopf algebra H on an m-Koszul Artin–Schelter regular algebra A. Such an algebra A is a derivation-quotient algebra for some twisted superpotential w, and we show that the homological determinant of the action of H on A can be easily calculated using w. Using this, we show that the smash product A#H is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which A#H is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.
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Acknowledgements
The author is a Heilbronn fellow at the University of Manchester. Portions of this work were completed at the University of Waterloo while the author was a postdoctoral fellow, and at the University of Washington while the author was in receipt of the Cecil King Travel Scholarship. The author is grateful for their financial support.
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Presented by: Henning Krause
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Crawford, S. Superpotentials and Quiver Algebras for Semisimple Hopf Actions. Algebr Represent Theor 26, 2709–2752 (2023). https://doi.org/10.1007/s10468-022-10165-y
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DOI: https://doi.org/10.1007/s10468-022-10165-y
Keywords
- Artin-Schelter regular algebras
- Hopf algebra action
- McKay quiver
- Homological determinant
- Auslander map
- Twisted superpotential