Augmented down-up algebras and uniform posets

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.

The attenuated space poset Aq(N,M)

“…We now interpret our results so far in terms of augmented down-up algebras [33]. Fix a nonzero τ ∈ C which is not a root of unity .…”

“…dual eigenvalue sequence) of A, A * . Let β, γ, γ * , ̺, ̺ * denote scalars in C. Then these scalars satisfy (33), (34) if and only if the following (i)-(v) hold:…”

Prime Spectra of Ambiskew Polynomial Rings

“…We next look at a class of algebras, introduced by Terwilliger and Worawannotai [16], to which Proposition 2.9 applies with t = 1.…”

“…, c t ] of R for some t ≥ 0. This general result will be applied, with t = 1 to show that the augmented down-up algebras of [16] have the property that every non-zero ideal has non-zero intersection with the centre which, for these algebras, is a polynomial algebra in two indeterminates.…”

Simple ambiskew polynomial rings

“…New examples of ambiskew polynomial rings have continued to emerge, including the down-up algebras of Benkart and Roby [6], the generalized down-up algebras of Cassidy and Shelton [7] and, most recently, the augmented down-up algebras of Terwilliger and Worawannatoi [35]. To accommodate the non-Noetherian down-up algebras, the definition of ambiskew polynomial ring was amended to allow α to be non-bijective and ρ to be zero but here we shall assume that α is bijective and ρ = 0.…”