Homological invariants of the super Jordan plane

SEBASTIÁN RECA; ANDREA SOLOTAR

Faro

Conferencia; Seminar on Rep. Theory and Rel. Areas; 2017

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 rarelyallows explicit calculations to be carried out.Nichols algebras are generalizations of symmetric algebras in the context of braided tensor categories. These are graded algebras, which first appeared in an article by Nichols in 1978, in which the author looked for examples of Hopf algebras. They are fundamental objects for the classification of pointed Hopfalgebras, as was shown by the work of Andruskiewitsch and Schneider.Heckenberger classified finite-dimensional Nichols algebras of diagonal type up to isomorphism. The classification separates the Nichols algebras in different classes: Nichols algebras of the Cartan type, essentially related to the finite quantum groups of Lusztig; Nichols algebras related to finite quantum supergroups and a third class related to countergradient Lie superalgebras. Later, Angiono described the definition relations of the Nichols algebras of the  list of Heckenberger. The problem of finite generation of the cohomology of a Hopf algera is related to Hochschild cohomology, since for any augmented algebra, the former graded space is isomorphic to a direct summmand ofthe latter.In a joint work with Sebastián Reca, we computed the Hochschild homology and cohomology of the super Jordan plane.This is the Nichols algebra B(V(1,2)),B(, whose Gelfand-Kirillov dimension is 2.The main results we obtained are the following.? We give explicit bases for the Hochschild homology and cohomology spaces.? We describe the cup product. ? We describe Lie algebra structure of the first cohomology space., which turns out to be isomorphic to a Lie subalgebra of the Virasoro algebra.