On the Faithfulness of a Family of Representations of the Singular Braid Monoid SMₙ

For n≥ 2, let Gₙ be a group and let ρ: Bₙ→ Gₙ be a representation of the braid group Bₙ. For a field K and a,b,c∈ K, Valerij G. Bardakov extend the representation ρ to a family of representations Φa,b,c:SMₙ → K[Gₙ] of the singular braid monoid SMₙ, where K[Gₙ] is the group algebra of Gₙ over K. In this paper, we study the faithfulness of the family of representations Φa,b,c in some cases. First, we find necessary and sufficient conditions of the families Φa,0,0, Φ0,b,0 and Φ0,0,c for all n≥ 2 to be unfaithful, where a,b,c ∈ K^*. Second, we consider the case $n=2$ and we find the nature of (Φa,b,c) if Φa,b,c is unfaithful. Moreover, we study the faithfulness of some families Φa,b,c in this case. Also, we find the shape of the possible elements in (Φa,b,c) for all n≥ 2 when Φa,b,c|SM₂ is unfaithful.