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On representations of the multi-virtual braid group M_kVB_n and the multi-welded braid group M_kWB_n
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On representations of the multi-virtual braid group M_kVB_n and the multi-welded braid group M_kWB_n
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This paper classifies complex homogeneous $2$-local representations of the multiple virtual braid group $M_kVB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, showing that such representations fall into exactly $2^{k+1}+1$ distinct types, out of which except three all are unfaithful. In addition, this paper investigates complex homogeneous $2$-local representations of the multiple welded braid group $M_kWB_n$ into $\mathrm{GL}_n(\mathbb{C})$ for $n\geq3$ and $k >1$, identifying $3 \cdot 2^{k-1} +1$ representations. Moreover, the article includes a construction of a non-local representation of $M_2WB_3$ that extends the known LKB representation of the braid group on $3$ strands, namely $B_3$, making a path towards constructing non-local representations of $M_kWB_n$ in general.
Forward citations
Cited by 3 Pith papers
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On Universal Virtual and Welded Braid Groups and Their Linear Representations
Universal virtual and welded braid groups are constructed to unify prior variants, with classifications of their complex homogeneous local representations and proofs of abelianization, perfect commutator, trivial cent...
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Multi-welded twin groups
Introduces multi-welded twin groups M_kWT_n, establishes quotient maps and structural properties including abelianization and perfect commutator subgroup for n≥5, and classifies homogeneous 2-local and 3-local represe...
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Presentations and Representations of the Multi-Virtual Twin Group and Associated Subgroups
The multi-virtual twin group M_kVT_n admits exactly eight distinct homogeneous 2-local representations into GL_n(C) for n >= 3; these are generally unfaithful but irreducible under explicit conditions, with induced no...
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