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 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 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 char(k) is not equal to $2$ and $H{H}^1(A)\cong$ $s{l}_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.

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