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

DOI10.1016/J.AIM.2012.04.016zbMATH Open1267.81209arXiv1011.6520OpenAlexW2963766270MaRDI QIDQ436143

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Publication date: 30 July 2012

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Abstract: We study quadratic algebras over a field $extbfk$. 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 $c_{xy}inextbfk$. 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 $n$.

Full work available at URL: https://arxiv.org/abs/1011.6520

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