CONICET | Buscador de Institutos y Recursos Humanos

Hochschild cohomology and its Gerstenhaber algebra structure are relevantinvariants: they are invariant by Morita equivalences, by tilting processesand by derived equivalences.The computation of these invariants requires a resolution of the algebraconsidered as a bimodule over itself. Of course, there is always a canonicalresolution 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.Nichols algebras are generalizations of symmetric algebras in the contextof braided tensor categories. These are graded algebras, which first appearedin an article by Nichols in 1978, in which the author looked for examples ofHopf algebras. They are fundamental objects for the classification of pointedHopf algebras, as was shown by the work of Andruskiewitsch and Schneider.Heckenberger classified finite-dimensional Nichols algebras of diagonaltype up to isomorphism. The classification separates the Nichols algebras indifferent classes: Nichols algebras of the Cartan type, essentially related to the finite quantum groups of Lusztig; Nichols algebras related to finite quantumsupergroups and a third class related to countergradient Lie superalgebras.Later, Angiono described the definition relations of the Nichols algebras ofthe list of Heckenberger.The problem of finite generation of the chomology of a Hopf algera is relatedto Hochschild cohomology, since for any augmented algebra, the formergraded space is isomorphic to a direct summmand of the latter.We computed the Hochschild homology and cohomology of $A=k/(x^2, y^2x-xy^2- xyx)$, called the super Jordan plane, when char(k) = 0 and k is algebraically closed. This is the Nichols algebra B(V(-1, 2)), whose Gelfand-Kirillov dimension is 2.