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

par Prof. Tatiana GATEVA-IVANOVA (IMI, Bulgarian Academy of Sciences, American Univ. in Bulgaria & IHÉS)

vendredi 13 juin 2014, 11:00 → 12:00 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

We study two classes of n-generated quadratic algebras over a field K. The first is the class C_n of all n-generated PBW algebras with polynomial growth and finite global dimension. We show that a PBW algebra A is in C_n iff its Hilbert series is H_A(z) = 1/(1-z)ⁿ. Furthermore each class C_n contains a unique (up to isomorphism) monomial algebra A = K ‹ x₁, … , x_n › / (x_j x_i | 1 ≤£ i < j £ n). The second is the class of n-generated quantum binomial algebras A, where the defining relations are nondegenerate square-free binomials xy - c_xy zt, with nonzero coefficients c_xy. Our main result shows that the following conditions are equivalent: (i) A is a Yang-Baxter algebra, that is the set of quadratic relations R defines canonically a solution of the Yang-Baxter equation. (ii) A is an Artin-Schelter regular PBW algebra. (iii) A is a PBW algebra with polynomial growth. (iv) A is a binomial skew polynomial ring. (v) The Koszul dual A! is a quantum Grassmann algebra.