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

Serdica Mathematical Journal (2004)

- Volume: 30, Issue: 2-3, page 431-470
- ISSN: 1310-6600

## Access Full Article

top## Abstract

top## How to cite

topGateva-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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.