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Preprints Archive: Abstract of IC2010073 (2010)

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

by Tatiana Gateva-Ivanova

Document info: Pages 24, Figures 3.

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 iff its Hilbert series is $H_A(z)= 1 /(1-z)^n$. A surprising amount can be said when the algebra $A$ has quantum binomial relations, that is the defining relations are binomials $xy-c_{xy}zt$, $c_{xy}\in \textbf{k}^{\times}$, which are square-free and nondegenerate. We prove that 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. This implies that 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 $n$.

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