Down–Up Algebras and Ambiskew Polynomial Rings

We show that the down–up algebras of G. Benkart (1998, in “Recent Progress in Algebra,” Contemporary Mathematics Vol. 224, Am. Math. Soc., Providence) and G. Benkart and T. Roby (1998, J. Algebra209, 305–344) lie in a certain class of iterated skew polynomial rings, called ambiskew polynomial rings, in two indeterminates x and y over a commutative ring B. In such rings, commutation of the indeterminates with elements of B involve the same endomorphism σ of B, but from different sides, that is, yb = σ(b)y and bx = xσ(b), and, for some scalar p, yx − pxy B. In previous studies of ambiskew polynomial rings, σ was required to be an automorphism but, in order to cover all down–up algebras, this requirement must be dropped. The Noetherian down–up algebras are those where σ is an automorphism and, in this case, we apply existing results on ambiskew polynomial rings to determine the finite-dimensional simple modules and the prime ideals. We adapt the methods underlying these results so as to apply to the non-Noetherian down–up algebras for which they reveal a surprisingly rich structure.