Conference abstracts
Session S10 - Homological Methods
July 29, 15:30 ~ 16:00
Hochschild homology and cohomology of down-up algebras
Andrea Solotar
Universidad de Buenos Aires- IMAS-Conicet, Argentina - asolotar@dm.uba.ar
We have computed Hochschild homology and cohomology of homogeneous down-up algebras in the generic case and in the Calabi-Yau case.
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 $K\langle{u,d}\rangle$ by the ideal generated by the relations \begin{equation} \label{eq:relationsdownup} \begin{split} d^{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). \end{split} \end{equation} 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 $\mathscr{U}(sl(2,K))$.
E. Kirkman, I. Musson and D. Passman proved that $A(\alpha,\beta,\gamma)$ is noetherian if and only it is a domain, which is tantamount to the fact that the subalgebra of $A(\alpha,\beta,\gamma)$ generated by $u d$ and $d u$ is a polynomial algebra in two indeterminates, 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 $K$ is algebraically closed, then the global dimension of $A(\alpha,\beta,\gamma)$ is always $3$. Moreover, Benkart and Roby also proved that the Gelfand-Kirillov dimension of a down-up algebra is $3$, independently of the parameters.
If $\gamma = 0$, the down-up algebra can be regarded as nonnegatively graded, where the degree of $u$ and $d$ is $1$. In this case, the algebra is $3$-Koszul and Artin-Schelter regular.
Joint work with Sergio Chouhy. IMAS-CONICET, Argentina and Estanislao Herscovich. Universidad de Buenos Aires and Institut Joseph Fourier, Grenoble, France.