Positive Energy Representations of the Loop Groups of Non Simply Connected Lie Groups

We classify and construct all irreducible positive energy representations of the loop group of a compact, connected and simple Lie group and show that they admit an intertwining action of Diff(S^{1}).Comment: Available from Springer Verlag at http://link.springer.de

Flat Connections and Quantum Groups

We review the Kohno-Drinfeld theorem as well as a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection D on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture when g=sl(n) and when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection D and Cherednik's generalisation of the KZ connection to finite reflection groups.Comment: 20 pages. To appear in the Proceedings of the 2000 Twente Conference on Lie Groups, in a special issue of Acta Applicandae Mathematica

Monodromy of the Casimir connection of a symmetrisable Kac-Moody algebra

Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group B_W of W on V. We then prove that this action is uniquely equivalent to the quantum Weyl group action of B_W on a quantum deformation of V, that is an integrable, category O-module V_h over the quantum group U_h(g) such that V_h/hV_h is isomorphic to V. This extends a result of the second author which is valid for g semisimple.Comment: One reference added. 48 page

Quasi-Coxeter categories and a relative Etingof-Kazhdan quantization functor

Let g be a symmetrizable Kac-Moody algebra and U_h(g) its quantized enveloping algebra. The quantum Weyl group operators of U_h(g) and the universal R-matrices of its Levi subalgebras endow U_h(g) with a natural quasi-Coxeter quasitriangular quasibialgebra structure which underlies the action of the braid group of g and Artin's braid groups on the tensor product of integrable, category O modules. We show that this structure can be transferred to the universal enveloping algebra Ug[[h]]. The proof relies on a modification of the Etingof-Kazhdan quantization functor, and yields an isomorphism between (appropriate completions of) U_h(g) and Ug[[h]] preserving a given chain of Levi subalgebras. We carry it out in the more general context of chains of Manin triples, and obtain in particular a relative version of the Etingof-Kazhdan functor with input a split pair of Lie bialgebras. Along the way, we develop the notion of quasi-Coxeter categories, which are to generalized braid groups what braided tensor categories are to Artin's braid groups. This leads to their succint description as a 2-functor from a 2-category whose morphisms are De Concini-Procesi associahedra. These results will be used in the sequel to this paper to give a monodromic description of the quantum Weyl group operators of an affine Kac-Moody algebra, extending the one obtained by the second author for a semisimple Lie algebra.Comment: 63 pages. Exposition in sections 1 and 4 improved. Material added: definition of a split pair of Lie bialgebras (sect. 5.2-5.5), 1-jet of the relative twist (5.20), PROP description of the Verma modules L_-,N*_+ (7.6), restriction to Levi subalgebras (8.4), D-structures on Kac-Moody algebras (9.1

Coxeter categories and quantum groups

We define the notion of braided Coxeter category, which is informally a tensor category carrying compatible, commuting actions of a generalised braid group B_W and Artin's braid groups B_n on the tensor powers of its objects. The data which defines the action of B_W bears a formal similarity to the associativity constraints in a monoidal category, but is related to the coherence of a family of fiber functors. We show that the quantum Weyl group operators of a quantised Kac-Moody algebra U_h(g), together with the universal R-matrices of its Levi subalgebras, give rise to a braided Coxeter structure on integrable, category O-modules for U_h(g). By relying on the 2-categorical extension of Etingof-Kazhdan quantisation obtained in arXiv:1610.09744, we then prove that this structure can be transferred to integrable, category O-representations of g. These results are used in arXiv:1512.03041 to give a monodromic description of the quantum Weyl group operators of U_h(g) which extends the one obtained by the second author for a semisimple Lie algebra.Comment: Substantial revision. Changes include 1) greatly expanded introduction 2) Section 8 split into the new sections 8 (Universal Algebras) and 9 (Universal pre-Coxeter Structures) 3) more thorough account of the PROPic nature of transferred structure (Sect. 10) 4) notion of Upsilon and a-strict pre-Coxeter structures (Sect. 3, 9 and 10). Final version. To appear in Selecta Math. 81 page