$\tau$-tilting theory and silting theory of skew group algebra extensions

TL;DR

This paper explores the relationship between $ au$-tilting and silting theories in skew group algebras, establishing poset isomorphisms and inheritance conditions, with applications to preprojective and mesh algebras.

Contribution

It introduces new poset isomorphisms linking support $ au$-tilting and silting modules over $ au$-tilting finite algebras and their skew group extensions, unifying previous results.

Findings

Poset isomorphisms between $G$-stable support $ au$-tilting modules over $ ext{Λ}$ and $A$.

Inheritance of $ au$-tilting finiteness and silting discreteness from $ ext{Λ}$ to $A$.

Explicit descriptions of support $ au$-tilting modules for preprojective algebras.

Abstract

Let $\Lambda$ be a finite dimensional algebra with an action by a finite group $G$ and $A:= \Lambda *G$ the skew group algebra. One of our main results asserts that the canonical restriction-induction adjoint pair of the skew group algebra extension $\Lambda \subset A$ induces a poset isomorphism between the poset of $G$-stable support $\tau$-tilting modules over $\Lambda$ and that of $(\!\!\!\mod G)$-stable support $\tau$-tilting modules over $A$. We also establish a similar poset isomorphism of posets of appropriate classes of silting complexes over $\Lambda$ and $A$. These two results generalize and unify preceding results by Huang-Zhang, Breaz-Marcus-Modoi and the second and the third authors. Moreover, we give a practical condition under which $\tau$-tilting finiteness and silting discreteness of $\Lambda$ are inherited to those of $A$. As applications we study $\tau$-tilting theory and silting theory of the (generalized) preprojective algebras and the folded mesh algebras. Among other things, we determine the posets of support $\tau$-tilting modules and of silting complexes over preprojective algebra $\Pi(\Bbb{L}_{n})$ of type $\Bbb{L}_{n}$.