Thomas Weber

Infinitesimal braidings are natural transformations in symmetric categories which can be used to construct braidings on the formal power series expansion of the category. Most prominently, for semisimple Lie algebras g this leads to a braiding on formal power series of Ug-modules which is equivalent to that of the corresponding Drinfeld-Jimbo quantum group. In this talk we propose a more general approach to infinitesimal braidings which applies to arbitrary braided categories. The motivating idea is to understand such an infinitesimal braiding as a first order deformation of a given braiding. We call such categories pre-Cartier, as they generalize previously studied Cartier categories. In case of quasitriangular bialgebra (co)modules we discuss the algebraic structure equivalent to an infinitesimal braiding. These are Hochschild 2-cocycles iff a deformed version of the quantum Yang-Baxter equation holds. We discuss several examples of infinitesimal braidings, particularly on q-deformed SL(N), Sweedler's Hopf algebra and via twisting. As main results we provide an infinitesimal FRT construction and Tannaka-Krein reconstruction theorem for pre-Cartier bialgebras. The former admits canonical non-trivial solutions and consequently induces infinitesimal braidings on several classes of quantum homogeneous spaces. The talk is based on a collaboration with Ardizzoni, Bottegoni and Sciandra.