We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.

We describe the generalized Casimir operators and their actions on the positive representations Pλ of the modular double of split real quantum groups Uqq (gR). We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters λ. We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of q, and we discuss explicit examples in the lower rank cases.

The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras: subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author.

No abstract uploaded!

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.