Classification of Spin-Chain Braid Representations

Abstract

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