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

2024/05/08 by Mohamad N. Nasser, Nasser, Mohamad N. · 1 citation
Computer Science · Mathematics · #20F36 #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings, Modules, and Algebras #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2405.04888

openalex publication_date 2024/05/08 · openalex created_date 2024/05/11 · openalex updated_date 2026/07/28

Abstract

For n≥ 2, let G_n be a group and let ρ: B_n→ G_n be a representation of the braid group B_n. For a field \mathbbK and a,b,c∈ \mathbbK, Bardakov, Chbili, and Kozlovskaya extend the representation ρ to a family of representations Φ_a,b,c:SM_n → \mathbbK[G_n] of the singular braid monoid SM_n, where \mathbbK[G_n] is the group algebra of G_n over \mathbbK. In this paper, we study the faithfulness of the family of representations Φ_a,b,c in some cases. First, we find necessary and sufficient conditions of the families Φ_a,0,0, Φ_0,b,0 and Φ_0,0,c for all n≥ 2 to be unfaithful, where a,b,c ∈ \mathbbK^*. Second, we consider the case n=2 and we find the nature of ker(Φ_a,b,c) if Φ_a,b,c is unfaithful. Moreover, we show that there exist some families Φ_a,b,c that have trivial kernel in the case n=2. Also, we find the shape of the possible elements in ker(Φ_a,b,c) for all n≥ 3 when the kernel of Φ_a,b,c|_SM2 is nontrivial.

Cited by

• Matrix representations of the twisted virtual braid group and its extensions

Related

• Extensions of braid group representations to the monoid of singular braids
• On the irreducibility of singular braid group representations
• A Faithful Representation of the Singular Braid Monoid on Three Strands
• Classification of Homogeneous Local Representations of the Singular Braid Monoid
• On extensions of the standard representation of the braid group to the singular braid group