Maximal tori in $HH^1$ and the fundamental group

TL;DR

This paper explores the relationship between maximal tori in the Hochschild cohomology Lie algebra of a finite dimensional algebra and the algebra's fundamental groups, revealing invariance properties and finiteness results.

Contribution

It establishes that all maximal tori in $HH^1(A)$ correspond to fundamental groups, extending previous work, and proves finiteness of monomial algebras within derived equivalence classes.

Findings

Maximal tori in $HH^1(A)$ are dual to fundamental groups.

Largest rank of a fundamental group is a derived invariant.

Finitely many monomial algebras exist in each derived equivalence class.

Abstract

We investigate maximal tori in the Hochschild cohomology Lie algebra $HH^1(A)$ of a finite dimensional algebra $A$, and their connection with the fundamental groups associated to presentations of $A$. We prove that every maximal torus in $HH^1(A)$ arises as the dual of some fundamental group of $A$, extending work of Farkas, Green and Marcos; de la Pe\~na and Saor\'in; and Le Meur. Combining this with known invariance results for Hochschild cohomology, we deduce that (in rough terms) the largest rank of a fundamental group of $A$ is a derived invariant quantity, and among self-injective algebras, an invariant under stable equivalences of Morita type. Using this we prove that there are only finitely many monomial algebras in any derived equivalence class of finite dimensional algebras; hitherto this was known only for very restricted classes of monomial algebras.