Braidings of Tensor Spaces

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