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

We give a formula for the universal R R -matrix of the quantized universal enveloping algebra U q ( g ) . U_q(\mathfrak {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 T_{w_0} on each representation V V , 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 is equal to the universal R R -matrix in finite type.