Mine is pretty standard. Denoting $\mathcal{U} = \mathcal{U}_q(\mathfrak{sl}_N)$ and $V$ the fundamental representation of $\mathcal{U}$, the braiding $\hat{R} : V \otimes V \to V \otimes V$ gives rise to a representation of the Hecke algebra $H_m(q)$ on $V^{\otimes m}$, which is the commutant of the representation of $\mathcal{U}$ on $V^{\otimes m}$. This gives a $q$-analogue of Schur-Weyl duality and tells you how to decompose $V^{\otimes m}$ as a direct sum of irreps of $\mathcal{U}$. For this perspective, one defines the Hecke algebra via generators and relations.