Let A be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of A is a simple directed graph, then HH 1 (A) is a solvable Lie algebra. The second main result shows that if the Ext-quiver of A has no loops and at most two parallel arrows in any direction, and if HH 1 (A) is a simple Lie algebra, then char(k) = 2 and HH 1 (A) ∼ = sl 2 (k). The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.

We show that if B is a block of a finite group algebra kG over an algebraically closed field k of prime characteristic p such that HH 1 (B) is a simple Lie algebra and such that B has a unique isomorphism class of simple modules, then B is nilpotent with an elementary abelian defect group P of order at least 3, and HH 1 (B) is in that case isomorphic to the Jacobson-Witt algebra HH 1 (kP ). In particular, no other simple modular Lie algebras arise as HH 1 (B) of a block B with a single isomorphism class of simple modules.2010 Mathematics Subject Classification. 16E40, 16G30, 16D90.

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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.

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