$\tau$-tilting finite triangular matrix algebras

TL;DR

This paper investigates conditions under which triangular matrix algebras over finite-dimensional algebras are $ au$-tilting finite and silting-discrete, providing new examples and classifications in the representation theory of algebras.

Contribution

It introduces new examples of silting-discrete algebras and classifies when triangular matrix algebras are $ au$-tilting finite.

Findings

Identified new silting-discrete algebras

Classified $ au$-tilting finite triangular matrix algebras

Explored conditions for silting-discreteness in triangular matrix algebras

Abstract

First, we give a new example of silting-discrete algebras. Second, one explores when the algebra of triangular matrices over a finite dimensional algebra is $\tau$-tilting finite. In particular, we classify algebras over which triangular matrix algebras are $\tau$-tilting finite. Finally, we investigate when a triangular matrix algebra is silting-discrete.