TY - JOUR

T1 - Representations of Flat Virtual Braids by Automorphisms of Free Group

AU - Chuzhinov, Bogdan

AU - Vesnin, Andrey

N1 - A.V.’s work was carried out in the framework of the Sobolev Institute of Mathematics project FWNF-2022-0004. Публикация для корректировки.

PY - 2023/8

Y1 - 2023/8

N2 - Representations of braid group (Formula presented.) on (Formula presented.) strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group (Formula presented.) on (Formula presented.) strands is an extension of the classical braid group (Formula presented.) by the symmetric group (Formula presented.). In this paper, we consider flat virtual braid groups (Formula presented.) on (Formula presented.) strands and construct a family of representations of (Formula presented.) by automorphisms of free groups of rank (Formula presented.). It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case (Formula presented.) and proved that the kernel contains a free group of rank two for (Formula presented.).

AB - Representations of braid group (Formula presented.) on (Formula presented.) strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group (Formula presented.) on (Formula presented.) strands is an extension of the classical braid group (Formula presented.) by the symmetric group (Formula presented.). In this paper, we consider flat virtual braid groups (Formula presented.) on (Formula presented.) strands and construct a family of representations of (Formula presented.) by automorphisms of free groups of rank (Formula presented.). It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case (Formula presented.) and proved that the kernel contains a free group of rank two for (Formula presented.).

KW - automorphism of free group

KW - braid

KW - flat virtual braid group

KW - virtual braid

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85168901087&origin=inward&txGid=2a3c9f437bf694b9c573dd032cda0a1c

UR - https://www.mendeley.com/catalogue/80c2950d-b886-3b70-8d56-d5633b300962/

U2 - 10.3390/sym15081538

DO - 10.3390/sym15081538

M3 - Article

VL - 15

JO - Symmetry

JF - Symmetry

SN - 2073-8994

IS - 8

M1 - 1538

ER -