Affine $G_2$ Centralizer Algebras

Abstract

We show that ${\mathrm{End}}_{\mathbf{U}}(V_{\lambda }\otimes V^{\otimes n})$ is generated by the affine braid group $A{B}_n$ where $\mathbf{U}=U_q\mathfrak{g}(G_2)$, $V$ is its 7-dimensional irreducible representation and $V_{\lambda }$ is an arbitrary irreducible representation.