INFINITESIMAL BRAIDINGS AND PRE-CARTIER BIALGEBRAS

We propose an approach to infinitesimal braidings which applies to arbitrary braided monoidal categories. The motivating idea is to understand an infinitesimal braiding as a first order deformation of a given braiding. We call braided monoidal categories endowed with an infinitesimal braiding ‘pre-Cartier’, because they generalize previously studied Cartier categories. It is the main goal of this talk to present the algebraic structure on coquasitriangular bialgebras which characterizes infinitesimal braidings on their categories of comodules. It turns out that this pre-Cartier bialgebra structure corresponds to Hochschild 2-cocycles which satisfy a deformed version of the quantum Yang-Baxter equation, while it gives rise to Hochschild 2-coboundaries in the Cartier cotriangular Hopf algebra framework. We discuss explicit examples on q-deformed GL(2) and Sweedler's Hopf algebra. As main results we provide an infinitesimal FRT construction and a Tannaka-Krein reconstruction theorem for pre-Cartier coquasitriangular bialgebras. The former admits canonical non-trivial solutions and thus induces non-trivial infinitesimal R-forms on all FRT bialgebras. The talk is based on a collaboration with Ardizzoni, Bottegoni and Sciandra.