homological invariants of the super Jordan plane and its relation with the Virasoro algebra

ANDREA SOLOTAR; SEBASTIÁN RECA

Natal

Congreso; Lie and Jordan algebras: its applications and representations VII; 2017

Universidade de sao Paulo

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.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 Hopf algebras, 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 chomology of a Hopf algera is related to Hochschild cohomology, since for any augmented algebra, the former graded space is isomorphic to a direct summmand of the latter.In a joint work with Sebasti´an Reca, we computed the Hochschild homology and cohomology of A = k<x, y>/(x^2, y^2x - xy^2 - xyx), called the super Jordan plane, when char(k) = 0 and | is algebraically closed. This is the Nichols algebra B(V(-1, 2)), 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. From this description we see that the isomorphism between H^{2p}(A,A) and H^{2p+2}(A,A), where p > 0, is given by the multiplication with an element of H^2(A,A), and similarly for the odd degrees. We describe Lie algebra structure of H^1(A,A), which turns out to be isomorphic to a Lie subalgebra of the Virasoro algebra. We describe the H^1(A,A)-module structure of the vector spaces H^n(A,A) for all n >2 and classify these representations.