The $\mathbb Z_3$-Symmetric Down-Up algebra

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra $\mathcal A$. The algebra $\mathcal A$ is associative, noncommutative, and infinite-dimensional. It is defined by two generators $A,B$ and two relations called the down-up relations. In the present paper, we introduce the $\mathbb Z_3$-symmetric down-up algebra $\mathbb A$. We define $\mathbb A$ by generators and relations. There are three generators $A,B,C$ and any two of these satisfy the down-up relations. We describe how $\mathbb A$ is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, the $\mathfrak{sl}_3$ loop algebra, the Kac-Moody Lie algebra $A^{(1)}_2$, the $q$-Weyl algebra, the quantized enveloping algebra $U_q(\mathfrak{sl}_2)$, and the quantized enveloping algebra $U_q (A^{(1)}_2)$. We give some open problems and conjectures.