Generalized down-up algebras revisited from a viewpoint of Gröbner basis theory

Abstract

The so called generalized down-up algebras are revisited from a viewpoint of Gröbner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided ), and by means of homogeneous Gröbner 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.