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 $K\cl{u,d}$ by the ideal generated by the relations

\begin{equation}

\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 $\mathscr{U}(sl(2,K))$.

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 showed that $A(\alpha,\beta,\gamma)$ is isomorphic to $A(\alpha',\beta',\gamma')$ if and only if the following conditions hold

\begin{equation}

\begin{split}

&\text{either } \alpha' = \alpha \text{ and } \beta' = \beta, \text{ or }

\alpha' = - \alpha^{-1} \beta \text{ and } \beta' = \beta^{-1}, \\

&\text{both } \gamma \text{ and } \gamma' \text{ are $0$ or different from $0$}.

\end{split}

\end{equation}

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 $K$ 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.