Extensions of Braid Group Representations to the Monoid of Singular Braids

Extensions of Braid Group Representations to the Monoid of Singular Braids Full article

Journal

Mediterranean Journal of Mathematics
ISSN: 1660-5446 , E-ISSN: 1660-5454

Output data

Year: 2024, Volume: 21, Number: 6, Article number : 180, Pages count : 21 DOI: 10.1007/s00009-024-02718-w

Tags

Braid group, monoid of singular braids, group of singular braids, representations, Artin representation, linear representations, Burau representation, Lawrence–Krammer–Bigelow representation.

Authors

Bardakov Valeriy G. ^1,2,3 , Chbili Nafaa ⁴ , Kozlovskaya Tatyana A. ³

Affiliations

1	Sobolev Institute of Mathematics
2	Novosibirsk State Agrarian University
3	Regional Scientific and Educational Mathematical Center of Tomsk State University
4	Department of Mathematical Sciences, College of Science, United Arab Emirates University

Funding (1)

Russian Science Foundation

24-21-00102

Given a representation φ:Bn→Gn of the braid group Bn, n≥2 into a group Gn, we are considering the problem of whether it is possible to extend this representation to a representation Φ:SMn→An, where SMn is the singular braid monoid and An is an associative algebra, in which the group of units contains Gn. We also investigate the possibility of extending the representation Φ:SMn→An to a representation Φ~:SBn→An of the singular braid group SBn. On the other hand, given two linear representations φ1,φ2:H→GLm(k) of a group H into a general linear group over a field k, we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of SBn which is an extension of the Lawrence–Krammer–Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence–Krammer–Bigelow representation.

Bardakov V.G. , Chbili N. , Kozlovskaya T.A.
Extensions of Braid Group Representations to the Monoid of Singular Braids
Mediterranean Journal of Mathematics. 2024. V.21. N6. 180 :1-21. DOI: 10.1007/s00009-024-02718-w WOS Scopus РИНЦ OpenAlex

Submitted:	Mar 14, 2024
Accepted:	Aug 7, 2024
Published print:	Sep 4, 2024
Published online:	Sep 4, 2024

Web of science:	WOS:001309340700001
Scopus:	2-s2.0-85203068765
Elibrary:	74391502
OpenAlex:	W4402229038

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