Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova.
type: Item
instance of: scholarly article
title: Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova.
zbMATH Open document ID: 1337.16028
published in: Journal of Pure and Applied Algebra
publication date: 2015-12-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1502.00790
review text: For a vector space \(V\) over a field \(K\), solutions to the quantum Yang-Baxter equation (QYBE) form a class of linear maps \(R\colon V\otimes V\to V\otimes V\), including the identity map. Many solutions are obtained as deformations of the identity map, hence \textit{V. G. Drinfel'd}'s 1990 [Lect. Notes Math. 1510, 1-8 (1992; Zbl 0765.17014)] suggestion to consider set-theoretic solutions, that is, solutions induced by linear extension from a map \(X\times X\to X\times X\), where \(X\) is a basis of \(V\). For finite \(X\), the non-degenerate unitary set-theoretic solutions are equivalent to a binary operation \(\cdot\) on \(X\) such that the left multiplications \(\sigma_x(y):=x\cdot y\) are invertible and the equation \[ (x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z) \] holds in \(X\). Such a set \((X;\cdot)\) is called a cycle set. Every cycle set \(X\) gives rise to a cycle set \(\sigma(X):=\{\sigma_x\mid x\in X\}\) with a well-defined operation \(\sigma_x\cdot\sigma_y:=\sigma_{x\cdot y}\), the retraction of \(X\). A cycle set satisfying \(x\cdot x=x\) is said to be square-free. Such (finite) cycle sets \(X\) are in one-to-one correspondence with an important class of non-commutative Noetherian regular \(K\)-algebras with basis \(X\). \textit{T. Gateva-Ivanova}'s ``strong conjecture'' [J. Math. Phys. 45, No. 10, 3828-3858 (2004; Zbl 1065.16037)] states that for any finite square-free cycle set \(X\), an iterated retraction \(\sigma^n(X)\) must be a singleton. The paper under review provides the first counterexample to this conjecture, a cycle set of order 8. This example is contained in a computer list of cycle sets up to order eight, obtained in connection with the paper of \textit{P. Etingof} et al. [Duke Math. J. 100, No. 2, 169-209 (1999; Zbl 0969.81030)] on set-theoretic solutions of the QYBE. Incidentally, it is the unique counterexample in this list. It can be obtained as an iterated ``double'' of the two-element square-free cycle set. From a give…
Mathematics Subject Classification ID: 16T25
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 6524664
zbMATH Keywords: quantum Yang-Baxter equation
zbMATH Keywords: set-theoretic solutions
zbMATH Keywords: square-free cycle sets
author: L. Vendramin
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
OpenAlex ID: W1793020813
Wikidata QID: Q123164423
arXiv ID: 1502.00790
cites work: From racks to pointed Hopf algebras
cites work: Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles
cites work: Involutive Yang-Baxter groups
cites work: Retractability of set theoretic solutions of the Yang-Baxter equation
cites work: Braces and the Yang-Baxter equation
cites work: Virtual Yang-Baxter cocycle invariants
cites work: Finite quotients of groups of I-type.
cites work: Coxeter-like groups for set-theoretic solutions of the Yang-Baxter equation.
cites work: On some unsolved problems in quantum group theory
cites work: Set-theoretical solutions to the quantum Yang-Baxter equation
cites work: A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation
cites work: Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity
cites work: Multipermutation solutions of the Yang-Baxter equation
cites work: Set-theoretic solutions of the Yang-Baxter equation, graphs and computations
cites work: Matched pairs approach to set theoretic solutions of the Yang-Baxter equation.
cites work: Quantum spaces associated to multipermutation solutions of level two
cites work: Semigroups of \(I\)-type
cites work: A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation
cites work: Braces, radical rings, and the quatum Yang-Baxter equation.
cites work: SEMIDIRECT PRODUCTS IN ALGEBRAIC LOGIC AND SOLUTIONS OF THE QUANTUM YANG–BAXTER EQUATION
DOI: 10.1016/J.JPAA.2015.10.018
Recommended article: Left non-degenerate set-theoretic solutions of the Yang–Baxter equation and dynamical extensions of q-cycle sets
Recommended article: Degenerate involutive set-theoretic solutions to the Yang-Baxter equation
Recommended article: Construction of quasi-linear left cycle sets
Recommended article: Simplicity of indecomposable set-theoretic solutions of the Yang-Baxter equation
Recommended article: A new family of set-theoretic solutions of the Yang–Baxter equation
Recommended article: A wells type exact sequence for non-degenerate unitary solutions of the Yang-Baxter equation
Recommended article: The classification of nondegenerate uniconnected cycle sets
Recommended article: A family of irretractable square-free solutions of the Yang-Baxter equation
Recommended article: Folding of set-theoretical solutions of the Yang-Baxter equation.
Recommended article: Q3607798
description: scientific article