On the Lie algebra structure of $H\!H^1(A)$ 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 $\operatorname {Ext}$-quiver of $A$ is a simple directed graph, then $H\!H^1(A)$ is a solvable Lie algebra. The second main result shows that if the $\operatorname {Ext}$-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $H\!H^1(A)$ is a simple Lie algebra, then $\operatorname {char}(k)\neq$ $2$ and $H\!H^1(A)\cong$ $\operatorname {\mathfrak {sl}}_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.