Departement de Mathematiques et Applications, Ecole Normale Superieure
45, rue d'Ulm, 75230 Paris Cedex 05

E-mail Marc.Rosso@ens.fr

Quantum groups and combinatorics on words

Abstract:

Let V be a finite dimensional linear space, with a fixed basis. The combinatorics of Lyndon words allows to construct remarquable bases of the tensor algebra of V, of the free Lie algebra on V or on the shuffle algebra on V.

The Borel parts of the quantized enveloping algebras associated with simple Lie algebras can be realized as sub-Hopf algebras of quantized shuffle algebras and the formalism of Lyndon bases can be adapted to this quantum situation to provide Poincare-Birkhoff-Witt type bases and the explicit formula for the universal R-matrix. Along the way, one provides an intrinsic definition of a "root" for the quantized enveloping algebras.