R-matrices for quantum affine algebras and Khovanov-Lauda-Rouquier algebras, I

TL;DR

This paper explores the relationship between R-matrices in quantum affine algebras and Khovanov-Lauda-Rouquier algebras, establishing a functorial connection that preserves algebraic structures in certain types.

Contribution

It introduces a functor from KLR algebra modules to quantum affine algebra modules, linking their convolution and tensor product structures, especially for types A, D, E.

Findings

The distribution of poles of normalized R-matrices yields KLR algebras.

The functor preserves convolution and tensor products.

Exactness of the functor is established for types A, D, E.

Abstract

Let us consider a finite set of pairs consisting of good $U'_q(g)$-modules and invertible elements. The distribution of poles of normalized R-matrices yields Khovanov-Lauda-Rouquier algebras We define a functor from the category of finite-dimensional modules over the KLR algebra to the category of finite-dimensional $U_q'(g)$-modules. We show that the functor sends convolution products to tensor products and is exact if the KLR albera is of type A, D, E.