A formula for the R-matrix using a system of weight preserving endomorphisms

TL;DR

This paper presents a new formula for the universal R-matrix of quantized universal enveloping algebras, utilizing weight preserving endomorphisms, applicable to all symmetrizable Kac-Moody algebras.

Contribution

It introduces a novel approach to constructing the R-matrix using weight preserving endomorphisms, extending applicability beyond finite type algebras.

Findings

Formula is well-defined for all symmetrizable Kac-Moody algebras.

Established equivalence with the universal R-matrix in finite type cases.

Provides a more general construction compared to previous methods.

Abstract

We give a formula for the universal R-matrix of the quantized universal enveloping algebra $U_q(\g).$ This is similar to a previous formula due to Kirillov-Reshetikhin and Levendorskii-Soibelman, except that where they use the action of the braid group element $T_{w_0}$ on each representation, we show that one can instead use a system of weight preserving endomorphisms. One advantage of our construction is that it is well defined for all symmetrizable Kac-Moody algebras. However we have only established that the result in equal to the universal R-matrix in finite type.