{
  "id": "2405.04888",
  "version": "v2",
  "published": "2024-05-08T08:37:38.000Z",
  "updated": "2024-08-03T04:20:40.000Z",
  "title": "On the Faithfulness of a Family of Representations of the Singular Braid Monoid $SM_n$",
  "authors": [
    "Mohamad N. Nasser"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-08-03T04:20:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F36"
    ],
    "keywords": [
      "singular braid monoid",
      "representation",
      "faithfulness",
      "trivial kernel",
      "group algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}