Geodesic
Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation
Serdica Mathematical Journal, Tome 30 (2004) no. 2-3, pp. 431-470

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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.
Keywords: Yang-Baxter Equation, Quadratic Algebras, Artin-Schelter Regular Rings, Quantum Groups 
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     author = {Gateva-Ivanova, Tatiana},
     title = {Binomial {Skew} {Polynomial} {Rings,} {Artin-Schelter} {Regularity,} and {Binomial} {Solutions} of the {Yang-Baxter} {Equation}},
     journal = {Serdica Mathematical Journal},
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Gateva-Ivanova, Tatiana. Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation. Serdica Mathematical Journal, Tome 30 (2004) no. 2-3, pp. 431-470. http://geodesic.mathdoc.fr/item/SMJ2_2004_30_2-3_a16/