The first Hochschild cohomology as a Lie algebra

• Published in: Quaestiones mathematicae Vol. 46; no. 9; pp. 1955 - 1980
• Main Authors: Rubio y Degrassi, Lleonard, Schroll, Sibylle, Solotar, Andrea
• Format: Journal Article
• Language: English
• Published: Grahamstown Taylor & Francis 02-09-2023; Taylor & Francis Ltd
• Subjects: Algebra; Gerstenhaber bracket; Group theory; Hochschild cohomology; Homology; Lie algebras; Lie groups; quantum complete intersections; solvable Lie algebras; symmetric algebras
• ISSN: 1607-3606; 1727-933X
• DOI: 10.2989/16073606.2022.2115424

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections.