Augmented down-up algebras and uniform posets - Terwilliger, Paul (Ars mathematica contemporanea, 2013)
Motivated by the structure of the uniform posets we introduce the notion of an augmented down-up (or ADU) algebra. We discuss how ADU algebras are related to the down-up algebras defined by Benkart and Roby. For each ADU algebra we give two presentations by generators and relations. We also display a ?$\mathbb{Z}$?-grading and a linear basis. In addition we show that the center is isomorphic to a polynomial algebra in two variables. We display seven families of uniform posets and show that each gives an ADU algebra module in a natural way. The main inspiration for the ADU algebra concept comes from the second authors thesis concerning a type of uniform poset constructed using a dual polar graph.
Keywords: down-up algebra; dual polar graph; dual polar space; uniform poset