Abstract. The MZV algebra is the graded algebra over Q generated by all multiple zeta values. The stable derivation algebra is a graded Lie algebra version of the Grothendieck-Teichmüller group. We shall show that there is a canonical surjective Q-linear map from the graded dual vector space of the stable derivation algebra over Q to the new-zeta space, the quotient space of the sub-vector space of the MZV algebra whose grade is greater than 2 by the square of the maximal ideal. As a corollary, we get an upper-bound for the dimension of the graded piece of the MZV algebra at each weight in terms of the corresponding dimension of the graded piece of the stable derivation algebra. If some standard conjectures by Y. Ihara and P. Deligne concerning the structure of the stable derivation algebra hold, this will become a bound conjectured in Zagier’s talk at 1st European Congress of Mathematics. Via the stable derivation algebra, we can compare the new-zeta space with the l-adic Galois image Lie algebra which is associated with so the Galois representation on the pro-l fundamental group of P1 − {0,1, ∞}.

Abstract. A bijective map r: X 2 − → X 2, where X = {x1, · · · , xn} is a finite set, is called a set-theoretic solution of the Yang-Baxter equation (YBE) if the braid relation r12r23r12 = r23r12r23 holds in X 3. A non-degenerate involutive solution (X, r) satisfying r(xx) = xx, for all x ∈ X, is called squarefree solution. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions (X, r) and the associated Yang-Baxter algebraic structures — the semigroup S(X, r), the group G(X, r) and the k- algebra A(k, X, r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a squarefree solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra A(k, X, r) is Poincaré-Birkhoff-Witt type algebra, with respect to some appropriate ordering of X. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler. 1.

Abstract. We interpret the five parameter family of Macdonald-Koornwinder polynomials as vector valued spherical functions on quantum Grassmannians.

Abstract. In this paper, we try to answer the following question: given a modular tensor category A with an action of a compact group G, is it possible to describe in a suitable sense the “quotient ” category A/G? We give a full answer in the case when A = Vec is the category of vector spaces; in this case,

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are “physically feasible.” We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.

Abstract. For any finite-dimensional Lie bialgebra g, we construct a bialgebra Au,v(g) over the ring C[u][[v]], which quantizes simultaneously the universal enveloping bialgebra U(g), the bialgebra dual to U(g ∗), and the symmetric bialgebra S(g). Following [Tur89], we call Au,v(g) a biquantization of S(g). We show that the bialgebra Au,v(g ∗ ) quantizing U(g ∗), U(g) ∗ , and S(g ∗ ) is essentially dual to the bialgebra obtained from Au,v(g) by exchanging u and v. Thus, Au,v(g) contains all information about the quantization of g. Our construction extends Etingof and Kazhdan’s one-variable quantization of U(g) [EK96].

Abstract. In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may then be applied to H ∗-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the “generating matrix ” formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products. 1.

... independently De Concini (unpublished), conjectured that the monodromy of the Casimir connection ∇C introduced in [MTL] is described by Lusztig’s quantum Weyl group operators. This conjecture was proved in [TL1] for all representations of the Lie algebra g = sln and in [TL2] for a number of pairs (g, V) including vector and spin representations of classical Lie algebras and the adjoint representation of all complex, simple Lie algebras. The aim of this paper, and of its sequel [TL4] is to prove this conjecture for all g. Our strategy is inspired by Drinfeld’s proof of the equivalence of the monodromy of the Knizhnik–Zamolodchikov equations for g and the R–matrix representations coming from the quantum group U�g. It relies on the use of quasi–Coxeter algebras, which are to the generalised braid group of type g what Drinfeld’s quasitriangular quasibialgebras are to Artin’s braid groups Bn. Using this notion, and the associated deformation cohomology, which we call Dynkin diagram

Abstract. We show that all finite dimensional pointed Hopf algebras with the same diagram in the classification scheme of Andruskiewitsch and Schneider are cocycle deformations of each other. This is done by giving first a suitable characterization of such Hopf algebras, which allows for the application of a result of Masuoka about Morita-Takeuchi equivalence and of Schauenburg about Hopf Galois extensions. The “infinitesimal ” part of the deforming cocycle and of the deformation determine the deformed multiplication and can be described explicitly in terms of Hochschild cohomology. Applications to, and results for copointed Hopf algebras are also considered. Finite dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero, particularly when the group of points is abelian, have been studied quite extensively with various methods in [AS, BDG, Gr1, Mu]. The most far reaching results as yet in this area have been obtained in [AS], where a large

Dedicated to Professor R.P. Bambah on his 75th birthday Multiple polylogarithms in a single variable are defined by