Maximal Tori in HH1 and the Fundamental Group

Abstract

We investigate maximal tori in the Hochschild cohomology Lie algebra ${\operatorname {HH}}^1(A)$ 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 ${\operatorname {HH}}^1(A)$ arises as the dual of some fundamental group of $A$, extending the work by Farkas, Green, and Marcos; de la Peña and Saorín; 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.

On the Lie algebra structure of integrable derivations

“…The first Betti number β(Q) of the underying graph of N p−1 p−1 is 1. This is because the Gabriel quiver is connected, the number of edges is p− 1, the number of vertices is p − 1 and consequently [4] we have that the maximal toral rank is 1 and it is easy to see that the map f sending a 1 to a 1 and sending any other arrow to zero is a diagonal outer derivation. From Corollary 5, we have dim k (HH 1 (kS p )) = 1.…”

On the Lie algebra structure of integrable derivations

“…This is because the Gabriel quiver is connected, the number of edges is $$p-1$$, the number of vertices is $$p-1$$ and consequently $$\beta (Q)=(p-1)-(p-1)+1=1$$. As $$N^{p-1}_{p-1}$$ is monomial, by [6, Theorem C] we have that the maximal total rank is 1 and it is easy to see that the map $$f$$ sending $$a_1$$ to $$a_1$$ and sending any other arrow to zero is a diagonal outer derivation. From Corollary 4, we have $$\mathrm{dim}_k({\operatorname{HH}}^1(kS_p))=1$$.…”

Derived equivalence classification of Brauer graph algebras

“…Note that maximal tori in Out 0 ( ) are related to the maximal tori in the Hochschild cohomology Lie algebra HH 1 ( ) (see [BRyD23]).…”