Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity

By Tatiana Gateva-Ivanova. 2010

86 citations recorded by the source corpus.

We study quadratic algebras over a field $\textbf{k}$. We show that an $n$-generated PBW algebra $A$ has finite global dimension and polynomial growth \emph{iff} its Hilbert series is $H_A(z)= 1 /(1-z)^n$. Surprising amount can be said when the algebra $A$ has \emph{quantum binomial relations}, that is the defining relations are nondegenerate square-free binomials $xy-c_{xy}zt$ with non-zero coefficients $c_{xy}\in \textbf{k}$. In this case various good algebraic and homological properties are closely related. The main result shows that for an $n$-generated quantum binomial algebra $A$ the following conditions are equivalent: (i) A is a PBW algebra with finite global dimension; (ii) A is PBW and has polynomial growth; (iii) A is an Artin-Schelter regular PBW algebra; (iv) $A$ is a Yang-Baxter algebra; (v) $H_A(z)= 1/(1-z)^n;$ (vi) The dual $A^{!}$ is a quantum Grassman algebra; (vii) A is a binomial skew polynomial ring. So for quantum binomial algebras the problem of classification of Artin-Schelter regular PBW algebras of global dimension $n$ is equivalent to the classification of square-free set-theoretic solutions of the Yang-Baxter equation $(X,r)$, on sets $X$ of order $n$.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/819d3609290f3b5e0cbdfa5bfb5984e54ba4aba1

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In Algebra and Number Theory

  1. Agata Smoktunowicz
  2. Leandro Vendramin
  3. Victoria Lebed
  4. Ferran Cedó

In Geometry and Topology

  1. Agata Smoktunowicz
  2. Leandro Vendramin
  3. Victoria Lebed
  4. Ferran Cedó

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