On the Faithfulness of a Family of Representations of the Singular Braid Monoid $S{M}_n$

For $n\ge 2$, let $G_n$ be a group and let $\rho :B_n\to G_n$ 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 ${\mathrm{\Phi }}_{a,b,c}:S{M}_n\to \mathbb{K}[G_n]$ of the singular braid monoid $S{M}_n$, 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 ${\mathrm{\Phi }}_{a,b,c}$ in some cases. First, we find necessary and sufficient conditions of the families ${\mathrm{\Phi }}_{a,0,0},{\mathrm{\Phi }}_{0,b,0}$ and ${\mathrm{\Phi }}_{0,0,c}$ for all $n\ge 2$ to be unfaithful, where $a,b,c\in {\mathbb{K}}^{\ast }$. Second, we consider the case $n=2$ and we find the nature of $\ker ({\mathrm{\Phi }}_{a,b,c})$ if ${\mathrm{\Phi }}_{a,b,c}$ is unfaithful. Moreover, we show that there exist some families ${\mathrm{\Phi }}_{a,b,c}$ that have trivial kernel in the case $n=2$. Also, we find the shape of the possible elements in $\ker ({\mathrm{\Phi }}_{a,b,c})$ for all $n\ge 3$ when the kernel of ${{\mathrm{\Phi }}_{a,b,c}\mathrm{\mid }}_{S{M}_2}$ is nontrivial.