Isomorphisms of non noetherian down-up algebras

TL;DR

This paper solves the isomorphism problem for non noetherian down-up algebras by analyzing their quotients and homological invariants, establishing uniqueness of certain monomial algebras within the family.

Contribution

It introduces a method to determine isomorphisms of non noetherian down-up algebras using quotients and homological invariants, identifying unique monomial algebra cases.

Findings

No other down-up algebra is isomorphic to the monomial algebra A(0,0,0).

The monomial algebra A(0,0,0) is unique within the down-up algebra family.

Homological invariants and abelianization are effective tools for isomorphism classification.

Abstract

We solve the isomorphism problem for non noetherian down-up algebras $A(\alpha,0,\gamma)$ by lifting isomorphisms between some of their non commutative quotients. The quotients we consider are either quantum polynomial algebras in two variables for $\gamma = 0$ or quantum versions of the Weyl algebra $A_1$ for non zero $\gamma$. In particular we obtain that no other down-up algebra is isomorphic to the monomial algebra $A(0,0,0)$. We prove in the second part of the article that this is the only monomial algebra within the family of down-up algebras. Our method uses homological invariants that determine the shape of the possible quivers and we apply the abelianization functor to complete the proof.