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

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.