Universal $R$-operator for split real quantum groups

The universal $R$ operator for the positive representations of split real quantum groups is computed, generalizing the formula of compact quantum groups $U_q(g)$ by Kirillov-Reshetikhin and the formula in the case of $U_q(sl(2,R))$ by Faddeev and Teschner. The quantum Weyl element and Lusztig’s isomorphism in the positive setting will be used in the construction. We also introduce a $C*$-algebraic version of the split real quantum group in the language of multiplier Hopf algebras, so that the $R$ operator can be realized as the canonical element of certain Drinfeld’s Double construction.