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 ${\mathfrak{s}\mathfrak{l}}_2$ and ${\mathfrak{s}\mathfrak{l}}_3$, the ${\mathfrak{s}\mathfrak{l}}_3$ loop algebra, the Kac-Moody Lie algebra $A_2^{(1)}$, the $q$-Weyl algebra, the quantized enveloping algebra $U_q({\mathfrak{s}\mathfrak{l}}_2)$, and the quantized enveloping algebra $U_q(A_2^{(1)})$. We give some open problems and conjectures.