On the Faithfulness of a Family of Representations of the Singular Braid Monoid $SM_n$

TL;DR

This paper investigates the faithfulness of a family of representations of the singular braid monoid, identifying conditions for unfaithfulness and characterizing kernels in specific cases.

Contribution

It provides necessary and sufficient conditions for faithfulness of certain representations and analyzes the kernel structure for the singular braid monoid.

Findings

Certain representations are unfaithful under specific parameter conditions.

Existence of representations with trivial kernel for n=2.

Characterization of kernel elements for n≥3 when the kernel is nontrivial.

Abstract

For $n\geq 2$, let $G_n$ be a group and let $\rho: B_n\rightarrow G_n$ be a representation of the braid group $B_n$. For a field $\mathbb{K}$ and $a,b,c\in \mathbb{K}$, Bardakov, Chbili, and Kozlovskaya extend the representation $\rho$ to a family of representations $\Phi_{a,b,c}:SM_n \rightarrow \mathbb{K}[G_n]$ of the singular braid monoid $SM_n$, where $\mathbb{K}[G_n]$ is the group algebra of $G_n$ over $\mathbb{K}$. In this paper, we study the faithfulness of the family of representations $\Phi_{a,b,c}$ in some cases. First, we find necessary and sufficient conditions of the families $\Phi_{a,0,0}, \Phi_{0,b,0}$ and $\Phi_{0,0,c}$ for all $n\geq 2$ to be unfaithful, where $a,b,c \in \mathbb{K}^*$. Second, we consider the case $n=2$ and we find the nature of $\ker(\Phi_{a,b,c})$ if $\Phi_{a,b,c}$ is unfaithful. Moreover, we show that there exist some families $\Phi_{a,b,c}$ that have trivial kernel in the case $n=2$. Also, we find the shape of the possible elements in $\ker(\Phi_{a,b,c})$ for all $n\geq 3$ when the kernel of ${\Phi_{a,b,c}|}_{SM_2}$ is nontrivial.