Positive Representations of Split Real Quantum Groups: The Universal R Operator

The universal R operator for the positive representations of split real quantum groups is computed, generalizing the formula of compact quantum groups $\mathcal U_q(\mathfrak g)$ by Kirillov–Reshetikhin and Levendorskiĭ–Soibelman, and the formula in the case of $\mathcal U_{q\tilde q}(\mathfrak {sl}(2,\ \mathbb R))$ by Faddeev, Kashaev, and Bytsko-Teschner. Several new functional relations of the quantum dilogarithm are obtained, generalizing the quantum exponential relations and the pentagon relations. The quantum Weyl element and Lusztig's isomorphism in the positive setting are also studied in detail. Finally, we introduce a C*-algebraic version of the split real quantum group in the language of multiplier Hopf algebras, and consequently the definition of R is made rigorous as the canonical element of the Drinfeld's double U of certain multiplier Hopf algebra Ub. Moreover, a ribbon structure is introduced for an extension of U.