On the representations of the braid group and the welded braid group

Rana S. Kahil\(^a\), Mohammad N. Abdulrahim\(^{b*}\)

\(^a\)Department of Mathematics, Beirut Arab University, P.O. Box 11-5020, Beirut, Lebanon.

\(^b\)Department of Mathematics, Beirut Arab University, P.O. Box 11-5020, Beirut, Lebanon.

Correspondence should be addressed to: : rsk349@student.bau.edu.lb, , mna@bau.edu.lb

Copyright © 2023 Rana S. Kahil, Mohammad N. Abdulrahim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: : 22 Nov 2022; Accepted: 08 May 2023; Published Online: : 08 June 2023.

Abstract

We study the composition of F. R. Cohen’s map B_n → B_nk with the standard representation of B_nk, where B_n is the braid group on n strings. We prove that the obtained representation of B_n is isomorphic to the direct sum of k copies of the standard representation of B_n. A similar work is done for the welded braid group 𝓌B_n.

Keywords:

Braid group, common fixed point, Welded Braid group, Cohen