Generalized Down-up Algebras Revisited from A Viewpoint of Gr\"obner Basis Theory

TL;DR

This paper revisits generalized down-up algebras using Gr"obner basis theory, showing they are solvable polynomial algebras and exploring their graded structures through homogeneous Gr"obner relations.

Contribution

It explicitly proves that generalized down-up algebras are solvable polynomial algebras under certain conditions and analyzes their graded structures via Gr"obner basis methods.

Findings

Generalized down-up algebras are solvable polynomial algebras when λω ≠ 0.

The associated graded, Rees, and homogenized algebras are explicitly characterized.

Homogeneous Gr"obner defining relations are used to explore algebraic structures.

Abstract

The so called generalized down-up algebras are revisited from a viewpoint of Gr\"obner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided $\lambda\omega\ne 0$), and by means of homogeneous Gr\"obner defining relations, the associated graded structures of generalized down-up algebras, namely the associated graded algebras, Rees algebras, and the homogenized algebras of generalized down-up algebras, are explored comprehensively.