2,468 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
Development of binder system for manufacturing metallic and ceramic parts by Powder Injection Molding technology
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