On the Faithfulness of a Family of Representations of the Singular Braid Monoid $SM_n$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913457111040000 |
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| author | Nasser, Mohamad N. |
| author_facet | Nasser, Mohamad N. |
| contents | For $n\geq 2$, let $G_n$ be a group and let $ρ: B_n\rightarrow G_n$ 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 $ρ$ to a family of representations $Φ_{a,b,c}:SM_n \rightarrow \mathbb{K}[G_n]$ 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 $Φ_{a,b,c}$ 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 $n\geq 2$ to be unfaithful, where $a,b,c \in \mathbb{K}^*$. Second, we consider the case $n=2$ and we find the nature of $\ker(Φ_{a,b,c})$ 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 $n\geq 3$ when the kernel of ${Φ_{a,b,c}|}_{SM_2}$ is nontrivial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04888 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Faithfulness of a Family of Representations of the Singular Braid Monoid $SM_n$ Nasser, Mohamad N. Representation Theory 20F36 For $n\geq 2$, let $G_n$ be a group and let $ρ: B_n\rightarrow G_n$ 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 $ρ$ to a family of representations $Φ_{a,b,c}:SM_n \rightarrow \mathbb{K}[G_n]$ 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 $Φ_{a,b,c}$ 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 $n\geq 2$ to be unfaithful, where $a,b,c \in \mathbb{K}^*$. Second, we consider the case $n=2$ and we find the nature of $\ker(Φ_{a,b,c})$ 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 $n\geq 3$ when the kernel of ${Φ_{a,b,c}|}_{SM_2}$ is nontrivial. |
| title | On the Faithfulness of a Family of Representations of the Singular Braid Monoid $SM_n$ |
| topic | Representation Theory 20F36 |
| url | https://arxiv.org/abs/2405.04888 |