Brick-splitting Torsion Pairs and Left Modularity

TL;DR

This paper introduces brick-splitting torsion pairs as a generalization of classical torsion pairs, characterizes them via lattice properties like left modularity, and explores their implications for brick-directed algebras and their structures.

Contribution

It defines and characterizes brick-splitting torsion pairs, introduces brick-directed algebras, and provides new insights into their properties and classifications.

Findings

Brick-splitting torsion pairs are characterized by lattice-theoretical properties.

Brick-directed algebras are a new class generalizing representation-directed algebras.

Brick-finite algebras are brick-directed iff their torsion class lattice is left modular.

Abstract

We introduce the notion of brick-splitting torsion pairs as a modern analogue and generalization of the classical notion of splitting torsion pairs. A torsion pair is called brick-splitting if any given brick is either torsion or torsion-free with respect to that torsion pair. After giving some properties of these pairs, we fully characterize them in terms of some lattice-theoretical properties, including left modularity. This leads to the notion of brick-directed algebras, which are those for which there does not exist any cycle of non-zero non-isomorphisms between bricks. This class of algebras is a novel generalization of representation-directed algebras. We show that brick-directed algebras have many interesting properties and give several characterizations of them. In particular, we prove that a brick-finite algebra is brick-directed if and only if the lattice of torsion classes is left modular (or equivalently, extremal). We also give a characterization of brick-directed algebras in terms of their wall-and-chamber structure, as well as of a certain Newton polytope associated to them. Moreover, we introduce an explicit construction of an abundance of brick-directed algebras, both of the tame and wild representation types.