Elliptic R-matrices and Feigin and Odesskii’s elliptic algebras

• Alex Chirvasitu ^[2] ; Ryo Kanda ^[3] ; S. Paul Smith ^[1]
  1. [1] University of Washington

     University of Washington

     Estados Unidos

     
  2. [2] University at Buffalo, USA
  3. [3] Osaka Metropolitan University, Japan

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 2, 2023
• Idioma: inglés
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• Resumen

  • The algebras Qn,k(E,τ)introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers n>k≥1, a complex elliptic curve E, and a point τ ∈ E. The main result in this paper is that Qn,k(E,τ) has the same Hilbert series as the polynomial ring on n variables when τ is not a torsion point. We also show that Qn,k(E,τ) is a Koszul algebra, hence of global dimension n when τ is not a torsion point, and, for all but countably many τ, Qn,k(E,τ)is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining Qn,k(E,τ) is the image of an operator Rτ(τ) that belongs to a family of operators Rτ(z):Cn⊗Cn→Cn⊗Cn, z∈C, that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.