Projectively Wakamatsu Tilting Modules over One-Point Extensions

TL;DR

This paper develops a method to lift projectively Wakamatsu tilting modules from an algebra to its one-point extension, preserving mutation relations and classifying modules in specific cases.

Contribution

It introduces a new lifting technique for PWT modules over one-point extensions and provides a complete classification for certain algebra classes.

Findings

Lifting PWT modules preserves mutation relations under certain conditions.

Complete classification of PWT modules for source point extensions of representation-finite algebras.

Established a bijection between PWT modules over the extension and the original algebra.

Abstract

Let $\Gamma = \Lambda[M]$ be the one-point extension of an algebra $\Lambda$ by a $\Lambda$-module $M$. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from $\mathrm{mod}\,\Lambda$ to $\mathrm{mod}\,\Gamma$ by adding the new projective module, and prove that this lifting process perfectly preserves mutation relations under certain homological conditions. Furthermore, for source point extensions of representation-finite algebras, we obtain a complete classification of PWT $\Gamma$-modules in terms of those over $\Lambda$. In particular, we establish a bijection \[ \mathrm{PWT}(\Gamma) \longleftrightarrow \mathrm{PWT}(\Lambda) \coprod \mathrm{RPWT}(\Lambda, S_i). \] which yields the counting formula about $|\mathrm{PWT}(\Gamma)|$.