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Isomorphisms of Nonnoetherian Down-Up Algebras

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Abstract

We solve the isomorphism problem for nonnoetherian down-up algebras A(α, 0, γ) by lifting isomorphisms between some of their noncommutative quotients. The quotients we consider are either quantum polynomial algebras in two variables for γ = 0 or quantum versions of the Weyl algebra A 1 for nonzero γ. 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.

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Authors and Affiliations

  1. IMAS, UBA-CONICET, Consejo Nacional de Investigaciones Cientícas y Técnicas, Ciudad Universitaria, Pabellón I, 1428, Buenos Aires, Argentina

    Sergio Chouhy & Andrea Solotar

  2. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 1, 1428, Buenos Aires, Argentina

    Andrea Solotar

Authors
  1. Sergio Chouhy
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  2. Andrea Solotar
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Corresponding author

Correspondence to Sergio Chouhy.

Additional information

Presented by Paul Smith.

This work has been supported by the projects UBACYT 20020130100533BA, UBACYT 20020130200169BA, PIP-CONICET 11220150100483CO, and MATHAMSUD-REPHOMOL. The second author is a research member of CONICET (Argentina).

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Chouhy, S., Solotar, A. Isomorphisms of Nonnoetherian Down-Up Algebras. Algebr Represent Theor 21, 1343–1352 (2018). https://doi.org/10.1007/s10468-017-9749-1

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  • DOI: https://doi.org/10.1007/s10468-017-9749-1

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