Braid Group Actions and Tensor Products

TL;DR

This paper explores how braid group actions influence tensor products in quantum affine algebras, providing explicit conditions for cyclicity and identifying zeros of R-matrices, advancing understanding of their algebraic structure.

Contribution

It introduces a braid group action on imaginary roots and derives explicit cyclicity conditions for tensor products in quantum affine algebras.

Findings

Explicit braid group action on imaginary roots established

Cyclicity conditions for tensor products derived

Zeros of R-matrices identified at specific points

Abstract

We define an action of the braid group of a simple Lie algebra on the space of imaginary roots in the corresponding quantum affine algebra. We then use this action to determine an explicit condition for a tensor product of arbitrary irreducible finite--dimensional representations is cyclic. This allows us to determine the set of points at which the corresponding $R$--matrix has a zero.