TY - JOUR

T1 - Virtual and universal braid groups, their quotients and representations

AU - Bardakov, Valeriy

AU - Emel'Yanenkov, Ivan

AU - Ivanov, Maxim

AU - Kozlovskaya, Tatyana

AU - Nasybullov, Timur

AU - Vesnin, Andrei

N1 - The results were obtained during the program on knots and braid groups advised by V. Bardakov, T. Nasybullov and A. Vesnin in the frame of The First Workshop of the Mathematical Center in Akademgorodok. Publisher Copyright: © 2022 Walter de Gruyter GmbH, Berlin/Boston.

PY - 2022/7/1

Y1 - 2022/7/1

N2 - 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}.

AB - 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}.

UR - http://www.scopus.com/inward/record.url?scp=85126006495&partnerID=8YFLogxK

U2 - 10.1515/jgth-2021-0114

DO - 10.1515/jgth-2021-0114

M3 - Article

AN - SCOPUS:85126006495

VL - 25

SP - 679

EP - 712

JO - Journal of Group Theory

JF - Journal of Group Theory

SN - 1433-5883

IS - 4

ER -