Down-up Algebras
Georgia Benkart and Tom Roby
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 $U({\mathfrak s\mathfrak 1_2})$ of the Lie algebra ${\mathfrak s\mathfrak 1_2}$ 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 $\mathcal O$ 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.