Set-theoretic solutions of the Yang–Baxter equation, associated quadratic algebras and the minimality condition
| Autor: | Ferran Cedó, Eric Jespers, Jan Okniński |
|---|---|
| Přispěvatelé: | Algebra and Analysis, Mathematics, Algebra |
| Rok vydání: | 2020 |
| Předmět: |
Racks and quandles
Yang–Baxter equation General Mathematics 010102 general mathematics Dimension (graph theory) Structure (category theory) Graded ring Field (mathematics) Mathematics - Rings and Algebras 01 natural sciences Linear span 16S37 16S15 16T25 010101 applied mathematics Combinatorics Quadratic equation Rings and Algebras (math.RA) FOS: Mathematics 0101 mathematics Mathematics |
| Zdroj: | Revista Matemática Complutense. 34:99-129 |
| ISSN: | 1988-2807 1139-1138 |
| Popis: | Given a finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation and a field $K$, the structure $K$-algebra of $(X,r)$ is $A=A(K,X,r)=K\langle X\mid xy=uv \mbox{ whenever }r(x,y)=(u,v)\rangle$. Note that $A=\oplus_{n\geq 0} A_n$ is a graded algebra, where $A_n$ is the linear span of all the elements $x_1\cdots x_n$, for $x_1,\dots ,x_n\in X$. One of the known results asserts that the maximal possible value of $\dim (A_2)$ corresponds to involutive solutions and implies several deep and important properties of $A(K,X,r)$. Following recent ideas of Gateva-Ivanova \cite{GI2018}, we focus on the minimal possible values of the dimension of $A_2$. We determine lower bounds and completely classify solutions $(X,r)$ for which these bounds are attained in the general case and also in the square-free case. This is done in terms of the so called derived solution, introduced by Soloviev and closely related with racks and quandles. Several problems posed in \cite{GI2018} are solved. 23 pages |
| Databáze: | OpenAIRE |
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