Low-dimensional indecomposable representations of the braid group $B_3$

TL;DR

This paper classifies all indecomposable but reducible representations of the braid group B_3 in dimensions 2 and 3 over an algebraically closed field of characteristic zero, providing a complete and explicit description.

Contribution

It provides the first complete classification of low-dimensional indecomposable reducible representations of B_3, filling a gap in the understanding of braid group representations.

Findings

Complete classification of 2- and 3-dimensional indecomposable reducible representations of B_3

Explicit examples illustrating the utility of these classifications

Framework for understanding the structure of B_3 representations

Abstract

In this note we give a complete classification of all indecomposable yet reducible representations of $B_3$ for dimensions $2$ and $3$ over an algebraically closed field $K$ with characteristic $0$, up to equivalence. We illustrate their utility with an example.