Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

Gateva-Ivanova, Tatiana. "Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation." Serdica Mathematical Journal 30.2-3 (2004): 431-470. <http://eudml.org/doc/219550>.

@article{Gateva2004,
abstract = {2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.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. The author was partially supported by the Department of Mathematics of Harvard University, by Grant MM1106/2001 of the Bulgarian National Science Fund of the Ministry of Education and Science, and by the Abdus Salam International Centre for Theoretical Physics (ICTP).},
author = {Gateva-Ivanova, Tatiana},
journal = {Serdica Mathematical Journal},
keywords = {Yang-Baxter Equation; Quadratic Algebras; Artin-Schelter Regular Rings; Quantum Groups; Yang-Baxter equation; quadratic algebras; Artin-Schelter regular rings; quantum groups; semigroup algebras; skew polynomial rings},
language = {eng},
number = {2-3},
pages = {431-470},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation},
url = {http://eudml.org/doc/219550},
volume = {30},
year = {2004},
}

TY - JOUR
AU - Gateva-Ivanova, Tatiana
TI - Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation
JO - Serdica Mathematical Journal
PY - 2004
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 30
IS - 2-3
SP - 431
EP - 470
AB - 2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.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. The author was partially supported by the Department of Mathematics of Harvard University, by Grant MM1106/2001 of the Bulgarian National Science Fund of the Ministry of Education and Science, and by the Abdus Salam International Centre for Theoretical Physics (ICTP).
LA - eng
KW - Yang-Baxter Equation; Quadratic Algebras; Artin-Schelter Regular Rings; Quantum Groups; Yang-Baxter equation; quadratic algebras; Artin-Schelter regular rings; quantum groups; semigroup algebras; skew polynomial rings
UR - http://eudml.org/doc/219550
ER -