A half-twist type formula for the R-matrix of a symmetrizable Kac-Moody algebra

TL;DR

This paper extends a known formula for the universal R-matrix from finite type Lie algebras to all symmetrizable Kac-Moody algebras by replacing the action with a bar-linear endomorphism.

Contribution

It introduces a new half-twist formula for the R-matrix applicable to all symmetrizable Kac-Moody algebras, generalizing previous finite type results.

Findings

Derived a new formula for the R-matrix in Kac-Moody algebras

Replaced the action of X with a bar-linear endomorphism

Extended the applicability of the R-matrix formula beyond finite type

Abstract

Kirillov-Reshetikhin and Levendorskii-Soibelman developed a formula for the universal R-matrix of the form R=(X^{-1} \otimes X^{-1}) \Delta(X). The action of X on a representation V permutes weight spaces according to the longest element in the Weyl group, so is only defined in finite type. We give a similar formula which is valid for the quantized universal enveloping algebra of any symmetrizable Kac-Moody algebra. This is done by replacing the action of X on V with an endomorphism that preserves weight spaces, but which is bar-linear instead of linear.