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

Abstract

Let A A be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext \operatorname {Ext} -quiver of A A is a simple directed graph, then H H 1 ( A ) H\!H^1(A) is a solvable Lie algebra. The second main result shows that if the Ext \operatorname {Ext} -quiver of A A has no loops and at most two parallel arrows in any direction, and if H H 1 ( A ) H\!H^1(A) is a simple Lie algebra, then char ⁡ ( k ) ≠ \operatorname {char}(k)\neq 2 2 and H H 1 ( A ) ≅ H\!H^1(A)\cong s l 2 ⁡ ( k ) \operatorname {\mathfrak {sl}}_2(k) . The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.