Braidings of Tensor Spaces

TL;DR

This paper explores how solutions to the Yang-Baxter equation on a braided vector space extend naturally to its tensor algebra, establishing a functorial relationship between these solutions.

Contribution

It introduces a method to construct solutions of the Yang-Baxter equation on tensor spaces from braided vector spaces, demonstrating a functorial correspondence.

Findings

Constructs solutions $T( ext{R})$ on tensor spaces from braided vector spaces.

Shows the functorial nature of the correspondence between $ ext{R}$ and $T( ext{R})$.

Provides a framework for extending solutions to higher tensor powers.

Abstract

Let $V$ be a braided vector space, that is, a vector space together with a solution $\hat{R}\in {\text{End}}(V\otimes V)$ of the Yang--Baxter equation. Denote $T(V):=\bigoplus_k V^{\otimes k}$. We associate to $\hat{R}$ a solution $T(\hat{R})\in {\text{End}}(T(V)\otimes T(V))$ of the Yang--Baxter equation on the tensor space $T(V)$. The correspondence $\hat{R}\rightsquigarrow T(\hat{R})$ is functorial with respect to $V$.