Classification of spin-chain braid representations

TL;DR

This paper constructs and classifies all spin-chain braid representations, which are monoidal functors from a category related to the Yang-Baxter equation to charge-conserving matrices, advancing understanding of their structure.

Contribution

It provides a complete construction and classification of all spin-chain braid representations within a categorical framework.

Findings

All spin-chain braid representations are explicitly constructed.

A classification up to isomorphism is achieved.

The work connects braid representations with charge-conserving matrices.

Abstract

A braid representation is a monoidal functor from the braid category $\mathsf{B}$, for example given by a solution to the constant Yang-Baxter equation. Given a monoidal category $\mathsf{C}$ with $ob(\mathsf{C})=\mathbb{N}$, a rank-$N$ charge-conserving representation (or spin-chain representation) is a strict monoidal functor $F$ from $\mathsf{C}$ to the category $\mathrm{Match}^N$ of rank-$N$ charge-conserving matrices that is natural in the sense that $F(1)=1$}. In this work we construct all spin-chain braid representations, and classify up to suitable notions of isomorphism.