Extensions of Braid Group Representations to the Monoid of Singular Braids Full article
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Mediterranean Journal of Mathematics
ISSN: 1660-5446 , E-ISSN: 1660-5454 |
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| 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 |
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| Affiliations |
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Funding (1)
| 1 | Russian Science Foundation | 24-21-00102 |
Abstract:
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.
Cite:
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
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
Dates:
| Submitted: | Mar 14, 2024 |
| Accepted: | Aug 7, 2024 |
| Published print: | Sep 4, 2024 |
| Published online: | Sep 4, 2024 |
Identifiers:
| Web of science: | WOS:001309340700001 |
| Scopus: | 2-s2.0-85203068765 |
| Elibrary: | 74391502 |
| OpenAlex: | W4402229038 |