Isomorphisms of non noetherian down-up algebras
Sergio Chouhy, Andrea Solotar

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 by lifting isomorphisms between some of their non commutative quotients. The quotients we consider are either quantum polynomial algebras in two variables for or quantum versions of the Weyl algebra for non zero . In particular we obtain that no other down-up algebra is isomorphic to the monomial algebra . 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.
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