Quasi-Hereditary Skew Group Algebras
Anna Rodriguez Rasmussen
TL;DR
This work investigates how quasi-hereditary structures and exact Borel subalgebras behave under the skew group algebra construction $A*G$ for a finite group $G$ acting on a finite-dimensional algebra $A$ with $|G|$ not divisible by ${\rm char}(k)$. The authors introduce $G$-equivariant and $G$-stable orders to relate the simples, standard modules, and filtrations of $A$ and $A*G$, proving that $(A,\le_A)$ is quasi-hereditary if and only if $(A*G,\le_{A*G})$ is, under a natural compatibility. They further show that an exact Borel subalgebra $B\subseteq A$ invariant under $G$ corresponds to an exact Borel subalgebra $B*G\subseteq A*G$, preserving properties like normality and regularity. An explicit description of the simples of $A*G$ is provided, and the results are applied to Auslander algebras of Nakayama algebras, illustrating Morita equivalences and the behavior of exact Borel subalgebras in a concrete setting. The Nakayama example reveals that regularity may fail for $N\ge3$ while holding for $N=2$, highlighting nuanced interactions between group actions and quasi-hereditary structure in skew group contexts.
Abstract
Given an algebra and a finite group acting on it via automorphisms, a natural object of study is the associated skew group algebra. In this article, we study the relationship between quasi-hereditary structures on the original algebra and on the corresponding skew group algebra. Assuming a natural compatibility condition on the partial order, we show that the skew group algebra is quasi-hereditary if and only if the original algebra is. Moreover, we show that in this setting an exact Borel subalgebra of the original algebra which is invariant as a set under the group action gives rise to an exact Borel subalgebra of the skew group algebra, and that under this construction, properties such as normality and regularity of the exact Borel subalgebra are preserved.