Abstract: It is well known that a solution for the Yang–Baxter equation (YBE) or equivalently for the braid equation (BE) gives a representation of the braid group B n . This paper is devoted to the generalization of this result to the case of the virtual braid group, related to the theory of virtual knots and playing an important role in quantum field theories. In this paper we explore a connection between YBE and representations of the virtual braid group VB n . In particular, we show that any solution ( X , R ) for the YBE with invertible R defines a representation of the virtual pure braid group VP n , for any n ≥ 2 , into Aut ( X ⊗ n ) for linear solution and into Sym ( X n ) for set-theoretic solution. Any solution of the BE with invertible R gives a representation of a normal subgroup H n of VB n . As a consequence of these two results we get that any invertible solution for the BE or YBE gives a representation of VB n . We also elaborate the technique of the group Y n connected with the problem of extension of YBE solutions, that is the construction of the YBE solution on the direct product B × C by solutions on the factors ( B , R B ) and ( C , R C ) .

Cite: Bardakov V.G. , Talalaev D.V.
Yang–Baxter equation and representations of the virtual braid group
Journal of Algebra and its Applications. 2025. 2650213 :1-21. DOI: 10.1142/s0219498826502130 WOS Scopus OpenAlex

Dates:

Submitted:	Mar 8, 2023
Accepted:	Mar 27, 2025
Published print:	Apr 25, 2025
Published online:	Apr 25, 2025

Identifiers:

Web of science:	WOS:001477521100001
Scopus:	2-s2.0-105003831128
OpenAlex:	W4409897456

Citing: Пока нет цитирований

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