The Gerstenhaber bracket in Hochschild cohomology

ANDREA SOLOTAR

Ciudad de México

Congreso; Advances in Representation Theory VII (ARTA 2018); 2018

UNAM

Homological methods provide important information about the structure of associative algebras, revealing sometimes hidden connections amongst them. The Gerstenhaber bracket in Hochschild cohomology of unital associative algebras over a field is an invariant preserved by derived equivalences.There has been a significant amount of effort expended by many authors in order to study this structure, specially in recent times.In particular, the Gerstenhaber bracket provides the first Hochschild cohomology space of a Lie algebra structure. The computation of Hochschild cohomology requires a resolution of the algebra considered as a bimodule over itself. Of course, there is always a canonical resolution available, the bar resolution, very useful from a theoretical point of view, but not very satisfactory in practice: the complexity of this resolution rarely allows explicit calculations to be carried out. The use of alternative resolutions is not well adapted to the computation of the Gerstenhaber bracket. However, some results by Witherspoon?Negron, Volkov and Suárez Álvarez provide useful tools to solve this problem. I will illustrate how, using these methods, it is possible to describe the first Hochschild cohomology spacesof some families of algebras -both finite and infinite dimensional - as Lie algebras, and relate this to the structure of the algebras.