On the Lie algebra structure of HH1(A) of a finite-dimensional algebra A

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 HH1(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 HH1(A) is a simple Lie algebra, then char(k)= 2 and HH1(A) ∼= sl2(k). The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.