The universal Teichmu¨ller space is an infinitely dimensional generalization of the classical Teichmu¨ller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil-Petersson metric. In this paper we investigate the Weil-Petersson Riemannian curvature operator ˜ Q of the universal Teichmu¨ller space with the Hilbert structure, and prove the following: (i) ˜ Q is non-positive definite. (ii) ˜ Q is a bounded operator. (iii) ˜ Q is not compact; the set of the spectra of ˜ Q is not discrete. As an application, we show that neither the Quaternionic hyperbolic space nor the Cayley plane can be totally geodesically immersed in the universal Teichmu¨ller space endowed with the Weil-Petersson metric.

The universal R operator for the positive representations of split real quantum groups is computed, generalizing the formula of compact quantum groups forumla by Kirillov–Reshetikhin and Levendorskiĭ–Soibelman, and the formula in the case of forumla 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.

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