Skew group algebras of Jacobian algebras
Simone Giovannini, Andrea Pasquali

TL;DR
This paper investigates skew group algebras of Jacobian algebras derived from quivers with potential under cyclic group actions, explicitly constructing potentials on the resulting quivers and exploring properties like self-injectivity and 2-representation finiteness.
Contribution
It provides an explicit construction of potentials on quivers after skew group algebra formation, extending the understanding of their structure and properties under group actions.
Findings
Explicit potential construction on the new quiver $Q_G$
Preservation of self-injectivity in the skew group algebra
Insights into cuts and 2-representation finiteness under the construction
Abstract
For a quiver with potential with an action of a finite cyclic group , we study the skew group algebra of the Jacobian algebra . By a result of Reiten and Riedtmann, the quiver of a basic algebra Morita equivalent to is known. Under some assumptions on the action of , we explicitly construct a potential on such that . The original quiver with potential can then be recovered by the skew group algebra construction with a natural action of the dual group of . If is self-injective, then is as well, and we investigate this case. Motivated by Herschend and Iyama's characterisation of 2-representation finite algebras, we study how cuts on behave with respect to our construction.
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