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

TL;DR

This paper explores the deep connections between quadratic algebras, Yang-Baxter solutions, and Artin-Schelter regularity, establishing equivalences that facilitate classification of these algebraic structures.

Contribution

It demonstrates that for quantum binomial algebras, several key properties are equivalent, linking algebraic, homological, and combinatorial classifications.

Findings

Equivalence of conditions for quantum binomial algebras with finite global dimension

Classification of Artin-Schelter regular PBW algebras via Yang-Baxter solutions

Characterization of dual algebras as quantum Grassman algebras

Abstract

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 $n$.