Maximal Tori in <i>HH</i>1 and the Fundamental Group

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