Hochschild cohomology of down-up algebras

SERGIO CHOUHY; E. HERSCOVICH; A. SOLOTAR

Porto

Seminario; Algebra, Combinatorics and Number Theory Seminar; 2016

Centro de Matematica - Universidade de Porto

Let K be a fixed field. Given parameters (\alpha,\beta,\gamma) in K^{3}, the associated down-up algebra A(\alpha,\beta,\gamma) is defined as the quotient of the free associative algebra K{u,d}$by the ideal generated by the relationsd^{2} u - (\alpha d u d + \beta u d^{2} + \gamma d),d u^{2} - (\alpha u d u + \beta u^{2} d + \gamma u).This family of algebras was introduced by G. Benkart and T. Roby. As typical examples we have that A(2,-1,0) is isomorphic to the enveloping algebra of the Heisenberg-Lie algebra of dimension 3, and, for $\gamma \neq 0$, A(2,-1,\gamma) is isomorphic to the enveloping algebra of $\mathfrak{sl}(2,K)$.  Moreover, Benkart proved that any down-up algebra such that $(\alpha, \beta) \neq (0,0)$ is isomorphic to one of Witten's 7-parameter deformations of $\mathscr{U}(sl(2,K))$. The down-up algebra $A(\alpha,\beta,\gamma)$ is isomorphic to $A(\alpha,\beta,1)$ for all $\gamma \neq 0$. Furthermore, if K is algebraically closed, P. Carvalho and I. Musson gave necessary and sufficient conditions for two down-up algebras to be isomorphic.E. Kirkman, I. Musson and D. Passman proved that $A(\alpha,\beta,\gamma)$ is noetherian if and only it is a domain, that in turn is equivalent to $\beta \neq 0$. Under either of the previous situations, $A(\alpha,\beta,\gamma)$ is Auslander regular and its global dimension is 3. On the other hand, it was proved by Cassidy and Shelton that, if $ is algebraically closed, then the global dimension of $A(\alpha,\beta,\gamma)$ is always 3. Moreover, Benkart and Roby proved that the Gelfand-Kirillov dimension of a down-up algebra is 3, independently of the parameters. Since $A(\alpha,\beta,\gamma)$ is isomorphic to the opposite algebra, left and right dimensions coincide.We have recently computed the Hochschild homology and cohomology of down-up algebras with $\gamma=0$ in the generic case and in the Calabi-Yau case ($\beta=-1$).In this talk I will report on these results. This is joint work with Sergio Chouhy and Estanislao Herscovich.