Dynamical Combinatorics and Torsion Classes

TL;DR

This paper explores the properties of the $ppa$-map within the lattice of torsion classes for artin algebras, extending its definition to infinite cases and relating it to known algebraic transformations.

Contribution

It extends the $ppa$-map to infinite torsion class lattices and connects it with Ringel's $psilon$-map and Auslander-Reiten translation in hereditary algebras.

Findings

The $ppa$-map is well-defined on infinite torsion class lattices.

Extended $ppa$-map coincides with Ringel's $psilon$-map in hereditary cases.

Square of $ppa$ relates to Auslander-Reiten translation.

Abstract

For finite semidistributive lattices the map $\kappa$ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the $\kappa$-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the $\kappa$-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend $\kappa$ to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll. For hereditary algebras, we show that the extended $\kappa$-map on torsion classes is essentially the same as Ringel's $\epsilon$-map on wide subcategories. Also in hereditary case, we relate the square of $\kappa$ to the Auslander-Reiten translation.