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Title: Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation
Authors: Gateva-Ivanova, Tatiana
Keywords: Yang-Baxter Equation
Quadratic Algebras
Artin-Schelter Regular Rings
Quantum Groups
Issue Date: 2004
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Mathematical Journal, Vol. 30, No 2-3, (2004), 431p-470p
Abstract: Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k \{0}, and z, t ∈ X is satisfied in A. We prove the equivalence of the notions a binomial skew polynomial ring and a binomial solution of YBE. This implies that the Yang-Baxter algebra of such a solution is of Poincaré-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.
Description: 2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.
ISSN: 1310-6600
Appears in Collections:Volume 30 Number 2-3

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