Down-up algebras were introduced by G. Benkart and T. Roby [J. Algebra 209, No. 1, 305-344 (1998; Zbl 0922.17006)], as generalizations of algebras generated by a pair of operators, the “down” and “up” operators, acting on the vector space \(\mathbb{C} P\) for certain partially ordered sets \(P\). Given a field \(K\) and \(\alpha,\beta,\gamma\in K\), the down-up algebra \(A(\alpha,\beta,\gamma)\) is the \(K\)-algebra generated by \(u\) and \(d\) subject to the relations \(d^2u=\alpha dud+\beta ud^2+\gamma d\) and \(du^2=\alpha udu+\beta u^2d+\gamma u\).
E. Kirkman, I. M. Musson and D. S. Passman [Proc. Am. Math. Soc. 127, No. 11, 3161-3167 (1999; Zbl 0940.16012)], have shown that \(A(\alpha,\beta,\gamma)\) is Noetherian if and only if \(\beta\neq 0\). In the present paper the authors determine the finite-dimensional simple modules over a Noetherian down-up algebra and, in certain cases, give a criterion for all finite-dimensional modules to be semisimple. There is some overlap here with the reviewer’s adjacent paper in the same volume [J. Algebra 228, No. 1, 311-346 (2000; Zbl 0958.16030)]. The authors also give a solution, complete in characteristic \(0\) and almost complete in finite characteristic, to the isomorphism problem for Noetherian down-up algebras. (Also submitted to MR).