The higher relation bimodule

TL;DR

This paper introduces the higher relation bimodule as a new construction for finite dimensional algebras, providing methods to analyze its quiver and demonstrating that the tensor algebra of this bimodule preserves gentleness in certain cases.

Contribution

It defines the higher relation bimodule for finite global dimension algebras and shows how to construct its quiver, proving gentleness preservation for gentle algebras.

Findings

Constructed quiver for the higher relation bimodule in string algebras

Proved tensor algebra of the higher relation bimodule is gentle for gentle algebras

Provided a recipe for the quiver construction in specific algebra classes

Abstract

Given a finite dimensional algebra A of finite global dimension, we consider the trivial extension of A by the A-A-bimodule $\oplus_{i\ge 2} \Ext^2_A(DA,A)$, which we call the higher relation bimodule. We first give a recipe allowing to construct the quiver of this trivial extension in case A is a string algebra and then apply it to prove that, if A is gentle, then the tensor algebra of the higher relation bimodule is gentle.