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Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity | Tatiana Gateva-Ivanova
; | Date: |
30 Nov 2010 | Abstract: | We study quadratic algebras over a field $ extbf{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 $c_{xy}in extbf{k}$. 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 $n$. | Source: | arXiv, 1011.6520 | Services: | Forum | Review | PDF | Favorites |
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