On the Lie Algebra Structure of HH¹(A) of a Finite-Dimensional Algebra A

Abstract

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) is not equal to 2 and HH¹(A)≅ sl2(k). The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.