For a natural number n, denote by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> the braid group on n strands. Y. Mikhalchishina classified all homogeneous 2-local representations of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. On the other hand, T. Mayassi and M. Nasser extended the result of Mikhalchishina by classifying all homogeneous 3-local representations of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n \ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. We consider in our work two group extensions of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. The first group, denoted by <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$VB_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msub> <mi>B</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, is the virtual braid group, and the second group, denoted by <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$WB_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <msub> <mi>B</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, is the welded braid group. Specifically, for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, Mikhalchishina constructed extensions of the Wada representations of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$VB_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msub> <mi>B</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$WB_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <msub> <mi>B</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. The Wada representations of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$B_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> are known to be local representations. As a generalization of the result of Mikhalchishina, we classify all homogeneous 2-local representations for all <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and all homogeneous 3-local representations for all <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> of both groups <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$VB_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msub> <mi>B</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$WB_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <msub> <mi>B</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. In addition, we study the faithfulness of these local representations in some cases.

Attacks on local representations of the virtual and the welded braid groups

期刊论文
发表日期：

2026年1月27日

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Mohamad N. Nasser

摘要

For a natural number n, denote by $B_n$ $B_{n}$ the braid group on n strands. Y. Mikhalchishina classified all homogeneous 2-local representations of $B_n$ $B_{n}$ for all $n \ge 3$ $n \geq 3$ . On the other hand, T. Mayassi and M. Nasser extended the result of Mikhalchishina by classifying all homogeneous 3-local representations of $B_n$ $B_{n}$ for all $n \ge 4$ $n \geq 4$ . We consider in our work two group extensions of $B_n$ $B_{n}$ . The first group, denoted by $VB_n$ $V B_{n}$ , is the virtual braid group, and the second group, denoted by $WB_n$ $W B_{n}$ , is the welded braid group. Specifically, for $n\ge 2$ $n \geq 2$ , Mikhalchishina constructed extensions of the Wada representations of $B_n$ $B_{n}$ to $VB_n$ $V B_{n}$ and $WB_n$ $W B_{n}$ . The Wada representations of $B_n$ $B_{n}$ are known to be local representations. As a generalization of the result of Mikhalchishina, we classify all homogeneous 2-local representations for all $n\ge 2$ $n \geq 2$ and all homogeneous 3-local representations for all $n\ge 4$ $n \geq 4$ of both groups $VB_n$ $V B_{n}$ and $WB_n$ $W B_{n}$ . In addition, we study the faithfulness of these local representations in some cases.