Irreducibility of the three dimensional Albeverio-Rabanovich representation of the pure braid group $P_3$

TL;DR

This paper proves that a specific three-dimensional linear representation of the pure braid group P_3, derived from the Albeverio-Rabanovich representation, is irreducible when specialized to non-zero complex numbers.

Contribution

It establishes the irreducibility of the specialized three-dimensional representation of P_3, providing new insights into the structure of these algebraic objects.

Findings

The representation remains irreducible after specialization.

Specialization to non-zero complex numbers preserves irreducibility.

The result clarifies the structure of the Albeverio-Rabanovich representation for P_3.

Abstract

We consider Albeverio- Rabanovich linear representation $\pi$ of the braid group $B_3$. After specializing the indeterminates used in defining the representation to non-zero complex numbers, we prove that the restriction of $\pi$ to the pure braid group $P_3$ of dimension three is irreducible.