  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.

Rings and Algebras