Augmented down-up algebras and uniform posets

TL;DR

This paper introduces augmented down-up (ADU) algebras inspired by uniform posets, providing algebraic structures, presentations, and connections to poset modules, expanding the understanding of algebra-poset relationships.

Contribution

It defines ADU algebras, relates them to down-up algebras, and demonstrates their application to seven families of uniform posets.

Findings

ADU algebras have two generator-relations presentations.

The center of ADU algebras is a polynomial algebra in two variables.

Seven families of uniform posets naturally give rise to ADU algebra modules.

Abstract

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 $Z$-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 author's thesis concerning a type of uniform poset constructed using a dual polar graph.