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

Abstract:

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