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

For n 2, let Gₙ be a group and let: 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 to a family of representations ₀, ₁, ₂: SMₙ KGₙ of the singular braid monoid SMₙ, where KGₙ is the group algebra of Gₙ over K. In this paper, we study the faithfulness of the family of representations ₀, ₁, ₂ in some cases. First, we find necessary and sufficient conditions of the families ₀, ₀, ₀, ₀, ₁, ₀ and ₀, ₀, ₂ 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 (₀, ₁, ₂) if ₀, ₁, ₂ is unfaithful. Moreover, we study the faithfulness of some families ₀, ₁, ₂ in this case. Also, we find the shape of the possible elements in (₀, ₁, ₂) for all n 2 when ₀, ₁, ₂|ₒ₌䃒 is unfaithful.