Abstract :
A down–up algebra A=A(α,β,γ), 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 d2u=αdud+βud2+γd, du2=αudu+βu2d+γu, where α, β, γ 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 . 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.
