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

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.

### 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

### Citations:

Zbl 1046.81054; Zbl 1432.16032; Zbl 1437.16028
Full Text:

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