Irreducible finite-dimensional modules of prime characteristic down-up algebras

A down-up algebra A = A (alpha, beta, gamma), as defined in a 1998 paper by Benkart and Roby [J. Algebra 209 (1998) 305-344; 213 (1999) 378 (Addendum)], is a unital associative algebra over a field K with two generators d and u and defining relations d(2)u = alphadud + betaud(2) + gammad, du(2) = alphaudu + betau(2)d + gammau, where alpha, beta, gamma are fixed scalars in K. This paper investigates the modules of down-up algebras over fields of characteristic p > 0. We start with the Verma modules and consider their weight spaces relative to h = Kdu circle plus Kud. We calculate exactly when a Verma module will break up into a finite number of infinite-dimensional weight spaces and when it splits into an infinite number of one-dimensional spaces. Using that result we then find all the finite-dimensional irreducible quotients of the Verma modules. Under the additional assumption that K is algebraically closed we determine all finite-dimensional irreducible modules for A, describing the actions of A on those modules and computing their dimension. (C) 2004 Elsevier Inc. All rights reserved.