Down–Up Algebras and Their Representations,

The down–up algebras were introduced in [G. Benkart and T. Roby, 1998, J. Algebra209, 305–344, and G. Benkart, 1998, Contemp. Math.224, 29–45]. Their representations were studied via category and Verma modules. Here we prove that when β ≠ 0 they are hyperbolic rings, and we study their representations via their left spectrum as defined in [A. L. Rosenberg, 1995, “Noncommutative Algebraic Geometry and Representations of Quantized Algebras,” Kluwer Academic, Dordrecht]. The center of these algebras is computed using results from hyperbolic rings which are also proved here. The case when the down–up algebra is a finite module over its center is considered in detail and its relation to the Clifford algebra of a binary form of arbitrary degree is established.