The [formula omitted]-symmetric down-up algebra

Abstract

In 1998, Georgia Benkart and Tom Roby introduced the down-up algebra A. The algebra 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 Z3-symmetric down-up algebra A. We define 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 A is related to some familiar algebras in the literature, such as the Weyl algebra, the Lie algebras sl2 and sl3, the sl3 loop algebra, the Kac-Moody Lie algebra A2(1), the q-Weyl algebra, the quantized enveloping algebra Uq(sl2), and the quantized enveloping algebra Uq(A2(1)). We give some open problems and conjectures.