DOWN-UP ALGEBRAS

GEORGIA BENKART AND TOM ROBY

Abstract. amsfonts The algebra generated by the down and up operators on a differential partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-up algebras. We show that down-up algebras exhibit many of the important features of the universal enveloping algebra U(𝔰𝔩_{2}) of the Lie algebra 𝔰𝔩_{2} including a Poincaré-Birkhoff-Witt type basis and a well-behaved representation theory. We investigate the structure and representations of down-up algebras and focus especially on Verma modules, highest weight representations, and category O modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets.