Marc Rosso
Paris-7
Quantum quasi-symmetric algebras and quantum groups.
Abstract: The "positive part" of quantum groups is known to be a quantum symmetric algebra (or Nichols algebra) associated with a particular braided vector space. From this point of view, one can realize the finite dimensional representations of the whole quantum group inside a larger quantum symmetric algebra, leading to new character formulas. We shall introduce quantum multibrace algebras and the particularly interesting and manageable subclass of quantum quasi symmetric algebras (roughly,"quasi" means that, compared to the quantum symmetric algebra situation, we have (and we use!) an extra algebra structure on the underlying braided vector space). We shall show that the whole quantum group is a (mild) quotient of a suitable quantum quasi symmetric algebra. This allows again to give a new realization of the representations, and also of the representations of the double of the quantum groups.