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September 10, 2025

Matrix Representation for Some Generalized Braid-Like Groups

Key Points

Generalized Burau representations are developed for virtual and welded braid groups, expanding current understanding.
The machinery provided allows for new welded pure braid representations, notably generalizing the classical Gassner representation.
The introduction of the virtual Iwahori–Hecke algebra connects classical frameworks to broader algebraic structures.
This work sheds light on relationships between virtual Temperley–Lieb and Iwahori–Hecke algebras, revealing new algebraic insights.

Abstract

In this paper, we construct generalized Burau representations of virtual and welded braid groups. For the Gassner representation, this paper gives a machinery for constructing new welded pure braid representations, and it allows to recover and generalize the extended classical Gassner representation. In addition, we introduce the virtual Iwahori–Hecke algebra with two parameters defined as an extension of the classical Iwahori–Hecke algebra and we observe that it has a connection with the virtual Temperley–Lieb algebra given by Paris and Rabenda.

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