A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation
Gateva-Ivanova, Tatiana (American University in Bulgaria)

Data:	2021
Resum:	We study noninvolutive set-theoretic solutions (X, r) of the Yang- Baxter equations in terms of the properties of the canonically associated braided monoid S(X, r), the quadratic Yang-Baxter algebra A = A(k, X, r) over a field k, and its Koszul dual A!. More generally, we continue our systematic study of nondegenerate quadratic sets (X, r) and their associated algebraic objects. Next we investigate the class of (noninvolutive) square-free solutions (X, r). This contains the self distributive solutions (quandles). We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We introduce and study a class of finite square-free braided sets (X, r) of order n ≥ 3 which satisfy the minimality condition, that is, dimk A2 = 2n − 1. Examples are some simple racks of prime order p. Finally, we discuss general extensions of solutions and introduce the notion of a generalized strong twisted union of braided sets. We prove that if (Z, r) is a nondegenerate 2-cancellative braided set splitting as a generalized strong twisted union of r-invariant subsets Z = X \∗ Y , then its braided monoid SZ is a generalized strong twisted union SZ = SX \ SY of the braided monoids SX and SY . We propose a construction of a generalized strong twisted union Z = X \ Y of braided sets (X, rX) and (Y, rY ), where the map r has a high, explicitly prescribed order.
Nota:	The author was partially supported by the Max Planck Institute for Mathematics (MPIM), Bonn (Fellowship 2019), by the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, and by Grant KP-06 N 32/1 of 07.12.2019 of the Bulgarian National Science Fund.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Yang-baxte ; Braided sets ; Quadratic sets ; Quadratic algebras
Publicat a:	Publicacions matemàtiques, Vol. 65 Núm. 2 (2021) , p. 747-808 (Articles) , ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/390250
DOI: 10.5565/PUBLMAT6522111

62 p, 603.6 KB

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