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.