# Braided fermions from Hurwitz algebras

## TL;DR

This paper explores the mathematical relationship between braid groups, Hurwitz algebras, and the internal symmetries of Standard Model fermions, proposing a novel algebraic framework for understanding fundamental particles.

## Contribution

It establishes a connection between braid groups, Hurwitz algebras, and Clifford algebras to model fermion states, offering a new algebraic perspective on particle symmetries.

## Key findings

- Braid groups $B_2$ and $B_3^c$ represent single-generation fermions.
- Fermion states correspond to basis states of minimal left ideals of $C
m ext{l}(6)$.
- Spectrum relates to octonion algebras and Clifford algebra structures.

## Abstract

Some curious structural similarities between a recent braid- and Hurwitz algebraic description of the unbroken internal symmetries for a single generations of Standard Model fermions were recently identified. The non-trivial braid groups that can be represented using the four normed division algebras are $B_2$ and $B_3^c$, exactly those required to represent a single generation of fermions in terms of simple three strand ribbon braids. These braided fermion states can be identified with the basis states of the minimal left ideals of the Clifford algebra $C\ell(6)$, generated from the nested left actions of the complex octonions $\mathbb{C}\otimes\mathbb{O}$ on itself. That is, the ribbon spectrum can be related to octonion algebras. Some speculative ideas relating to ongoing research that attempts to construct a unified theory based on braid groups and Hurwitz algebras are discussed.