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

2010/11/30 by Gateva-Ivanova, Tatiana · 2 citations
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1011.6520

Abstract

We study quadratic algebras over a field 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. Surprising amount can be said when the algebra A has quantum binomial relations, that is the defining relations are nondegenerate square-free binomials xy-c_xyzt with non-zero coefficients c_xy∈ 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.

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