75 research outputs found
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
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
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