Lie structure of the first Hochschild cohomology space

ANDREA SOLOTAR

Rio de Janeiro

Congreso; International Congress of Mathematicians; 2018

International Mathematical Uion

Hochschild cohomology and its Gerstenhaber algebra structure are relevant invariants: they are invariant by Morita equivalences, by tilting processes and by derived equivalences. The computation of these invariants 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.Recently, some important advances have been obtained in this direction.Here, we develop strategies well adapted to different types of algebras: the Jordan and the super Jordan plane (both of them are Nichols algebras, a family of special biserial algebras), and some subalgebras of the Weyl algebra. These strategies allow the complete computation of Gerstenhaber brackets, the description of the rst cohomology space as a Lie algebra and the Lie module structure of the higher cohomology spaces.The talk is based on results obtained in a joint work with Sebastian Reca, and work in progress with Samuel Lopes and with Joanna Meinel, Van Nguyen, Bregje Pauwels and Maria Julia Redondo.