2020 Thematic Program (I)

The Gerstenhaber bracket in the first Hochschild cohomology space


January 13-17, 2020

W303  School of Mathematics, Sichuan University


Andrea Solotar (U de Buenos Aires, Argentina)


Homological methods provide important information about the structure of associative algebras, revealing sometimes hidden connections amongst them. This course will be about an invariant preserved by derived equivalences: the Gerstenhaber bracket in the first Hochschild cohomology space of unital associative algebras over a field k. There has been a significant amount of effort expended by many authors in order to study this structure, especially in recent times. The first Hochschild cohomology space of an algebra A is the quotient of the k-linear derivations of A by the inner derivations, and the Gerstenhaber bracket provides it of a Lie algebra structure. Recent work in this area has been devoted to describe which Lie algebras appear in this way, and in particular to conditions on the algebra implying the solvability of HH^1(A) as Lie algebra. I will start by describing in detail HH^1(A), paying particular interest to its Lie structure, then treat some families of examples and finally give criteria on the algebra A to imply solvability of HH^1(A).