Lie Groups in Quasi-Poisson Geometry and Braided Hopf Algebras

Abstract

We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to $\mathfrak{g}$-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these $\mathfrak{g}$-quasi-Poisson Lie groups are nilpotent radicals of parabolic subgroups. We also provide examples of moment maps in this new context coming from moduli spaces of flat connections on surfaces.