The \(\mathbb{Z}_3\)-symmetric down-up algebra. (English) Zbl 07874687
Summary: 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 \(A_2^{(1)}\), the \(q\)-Weyl algebra, the quantized enveloping algebra \(U_q (\mathfrak{sl}_2)\), and the quantized enveloping algebra \(U_q (A_2^{(1)})\). We give some open problems and conjectures.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 15A21 | Canonical forms, reductions, classification |
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