Affine $G_2$ Centralizer Algebras

TL;DR

This paper demonstrates that the endomorphism algebra of certain tensor products in the quantum group of type G_2 is generated by the affine braid group, revealing new algebraic structure insights.

Contribution

It establishes that the affine braid group generates the endomorphism algebra for tensor products involving the 7-dimensional representation of U_q(g(G_2)).

Findings

Endomorphism algebra is generated by affine braid group.

Applicable to arbitrary irreducible representations.

Provides new understanding of quantum G_2 algebraic structures.

Abstract

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