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Cedó, Ferran; Jespers, Eric; Okniński, Jan

Set-theoretic solutions of the Yang-Baxter equation, associated quadratic algebras and the minimality condition. (English) Zbl 1466.16034

Rev. Mat. Complut. 34, No. 1, 99-129 (2021).
A braided set \((X,r)\) is a set \(X\) endowed with a set-theoretic solution \(r:X\times X\longrightarrow X\times X\) of the Yang-Baxter equation \[(r\times\operatorname{id})(\operatorname{id}\times r)(r\times\operatorname{id})=(\operatorname{id}\times r)(r\times\operatorname{id})(\operatorname{id}\times r).\] In the paper under review, the authors determine lower bounds for the numbers of orbits in \(X\times X\) under the action of \(r\).
Their first result asserts that \(X\times X\) has at least \(\frac{|X|}{2}\) (or \(\frac{|X|+1}{2}\), according to the parity of \(|X|\)) \(r\)-orbits if \(r(x,y)=(\sigma_x(y),x)\) with \(\sigma_x\) bijective for all \(x,y\in X\). Secondly, if \(r\) also satisfies \(r(x,x)=(x,x)\) for all \(x\in X\), then \(X\times X\) has at least \(2|X|-1\) \(r\)-orbits.
Furthermore, they explicitly describe the braided sets which reach the lower bound. In the first situation, these turn out to be those with \(\sigma_x=\sigma\), a cycle of length \(|X|\), for all \(x\in X\). In the second situation, the braided sets with minimal numbers of \(r\)-orbits are: (1) the set of reflections in the dihedral groups \(D_{2p}\) with \(p\) an odd prime; (2) the trivial braided set of order two; (3) \(X=\{1,2,3\}\) with \(\sigma_1=\sigma_2=\operatorname{id}\) and \(\sigma_3=(12)\).
The proofs are of combinatorial nature and accessible to non-experts.
Actually, the results of the authors are more general as the assumption \(r(x,y)=(\sigma_x(y),x)\) may be relaxed. Instead, one may assume that \(r(x,y)=(\sigma_x(y),\gamma_y(x))\) with \(\sigma_x\) and \(\gamma_y\) bijective for all \(x,y\in X\), and the lower bound still holds since the derived solution \((X,r')\) introduced by [A. Soloviev, Math. Res. Lett. 7, No. 5–6, 577–596 (2000; Zbl 1046.81054)] satisfies the first assumption and the numbers of \(r'\)-orbits and \(r\)-orbits are equals by [the second author et al., Trans. Am. Math. Soc. 372, No. 10, 7191–7223 (2019; Zbl 1432.16032)].
The authors also give some results and examples concerning the structure algebra of a braided set and several problems posed by T. Gateva-Ivanova [Adv. Math. 338, 649–701 (2018; Zbl 1437.16028)] are solved.
Reviewer: Cristian Vay (Córdoba)

Cited in 3 Documents

MSC:

16T25 Yang-Baxter equations
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
16S36 Ordinary and skew polynomial rings and semigroup rings
16S37 Quadratic and Koszul algebras
16W50 Graded rings and modules (associative rings and algebras)
16P90 Growth rate, Gelfand-Kirillov dimension
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory

Keywords:

Yang-Baxter; braided set; quadratic algebra; structure algebra; set-theoretic solution; dihedral quandle

Citations:

Zbl 1046.81054; Zbl 1432.16032; Zbl 1437.16028
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\textit{F. Cedó} et al., Rev. Mat. Complut. 34, No. 1, 99--129 (2021; Zbl 1466.16034)
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