On the Lie Algebra Structure of $HH^1(A)$ of a Finite-Dimensional Algebra A
MFO Scientific ProgramOWLF 2018
Rubio y Degrassi, Lleonard
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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) is not equal to $2$ and $HH^1(A)\cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.