On the Morita Reduced Versions of Skew Group Algebras of Path Algebras

Abstract

Let $R$ be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita-reduced algebra associated to $R$. Reiten and Riedtmann proved that there exists an idempotent $e$ of $R$ such that the algebra $eRe$ is both Morita equivalent to $R$ and isomorphic to the path algebra of some quiver, which was described by Demonet. This article gives explicit formulas for the decomposition of any element of $eRe$ as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category, which takes into account the representation theory of the finite group.

Orbifold diagrams

“…For convenience of the reader, we summarize here some points about this construction. If (Q, W ) is a QP and G is a finite group acting on Q fixing W , it is known by the work of Le Meur, [LM20], that the skew group algebra of the Jacobian algebra of (Q, W ) is Morita equivalent to the Jacobian algebra of a new QP (Q G , W G ). (We will use ∼ below to denote Morita equivalence).…”