Abstract: In the present paper, we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations B n → GL n (n - 1) / 2 (Z [ t ± 1 ]) B_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1}]), VB n → GL n (n - 1) / 2 (Z [ t ± 1, t 1 ± 1, t 2 ± 1, ..., t n - 1 ± 1 ]) \mathrm{VB}_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1},t_{1}^{\pm 1},t_{2}^{\pm 1},\ldots,t_{n-1}^{\pm 1}]) which are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations induce faithful representations of the crystallographic groups B n / P n ′ B_{n}/P_{n}^{\prime}, VB n / VP n ′ \mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime}, respectively. Using these representations we study certain properties of the groups B n / P n ′ B_{n}/P_{n}^{\prime}, VB n / VP n ′ \mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime}. Moreover, we construct new representations and decompositions of the universal braid groups UB n \mathrm{UB}_{n}.

Cite: Bardakov V. , Emel'Yanenkov I. , Ivanov M. , Kozlovskaya T. , Nasybullov T. , Vesnin A.
Virtual and universal braid groups, their quotients and representations
Journal of Group Theory. 2022. V.25. N4. P.679-712. DOI: 10.1515/jgth-2021-0114 WOS Scopus РИНЦ OpenAlex

Dates:

Published online:

Feb 25, 2022

Identifiers:

Web of science:	WOS:000761300700001
Scopus:	2-s2.0-85126006495
Elibrary:	48189695
OpenAlex:	W3178809773

Citing:

DB	Citing
Scopus	7
Web of science	7
OpenAlex	6
Elibrary	5

Altmetrics: