$The faithfulness of an extension of Lawrence-Krammer representation on the group of conjugating automorphisms $C_n$ in the cases $n=3$ and $n=4$$

The faithfulness of an extension of Lawrence-Krammer representation on the group of conjugating automorphisms $C_n$ in the cases $n=3$ and $n=4$

Let $C_n$ be the group of conjugating automorphisms. Valerij G. Bardakov defined a representation $\rho$ of $C_n$, which is an extension of Lawrence-Krammer representation of the braid group $B_n$. Bardakov proved that the representation $\rho$ is unfaithful for $n \geq 5$. The cases \(n...

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Veröffentlicht in:	arXiv.org 2023-08
1. Verfasser:	Nasser, Mohamad N
Format:	Artikel
Sprache:	eng
Schlagworte:	Automorphisms Braid theory Representations
Online-Zugang:	Volltext
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Zusammenfassung:	Let $C_n$ be the group of conjugating automorphisms. Valerij G. Bardakov defined a representation $\rho$ of $C_n$, which is an extension of Lawrence-Krammer representation of the braid group $B_n$. Bardakov proved that the representation $\rho$ is unfaithful for $n \geq 5$. The cases $n=3,4$ remain open. M. N. Nasser and M. N. Abdulrahim made attempts towards the faithfulness of $\rho$ in the case $n=3$. In this work, we prove that $\rho$ is unfaithful in the both cases $n=3$ and $n=4$.
ISSN:	2331-8422

Zusammenfassung:	Let \(C_n\) be the group of conjugating automorphisms. Valerij G. Bardakov defined a representation \(\rho\) of \(C_n\), which is an extension of Lawrence-Krammer representation of the braid group \(B_n\). Bardakov proved that the representation \(\rho\) is unfaithful for \(n \geq 5\). The cases \(n=3,4\) remain open. M. N. Nasser and M. N. Abdulrahim made attempts towards the faithfulness of \(\rho\) in the case \(n=3\). In this work, we prove that \(\rho\) is unfaithful in the both cases \(n=3\) and \(n=4\).
ISSN:	2331-8422

The faithfulness of an extension of Lawrence-Krammer representation on the group of conjugating automorphisms \(C_n\) in the cases \(n=3\) and \(n=4\)