KTH/SU Mathematics Colloquium

November 29, 2006

Alexander Stolin, Chalmers

Towards classification of quantum groups

ABSTRACT: In the fundamental paper "Quantum groups" V.Drinfeld has not defined explicitly the notion of a quantum group but he rather explained it on a number of examples. Later he posed the following question: Can any Lie bialgebra be quantized? A positive answer to the question was obtained by P.Etingof and D.Kazhdan. This answer justifies the following definition - 'A semi-classical quantum group is a quantization of a Lie bialgebra'. In a recent joint with S.Khoroshkin, A. Pop and V.Tolstoy paper we posed a conjecture, which refined Drinfeld's problem above. Namely, Can any classical twist be quantized? The latter problem was recently solved by G. Halbout in his paper "Formality theorem for Lie algebras and quantization of twists and coboundary r-matrices" (to appear in Adv. Math. December 2006). This result gives a basis for classification of quantum groups.