Reconstruction in braided categories and a notion of commutative bialgebra

Abstract

A braided group in the sense of Majid [6] is a Hopf algebra B in a braided monoidal category which satisfies a generalized commutativity condition; this condition is expressed with respect to a certain class of B-comodules. The more obvious condition that B be a commutative algebra in the braided category does not make sense. We propose a different commutativity condition for bialgebras: We show that a coalgebra reconstructed from a category over a braided base category %plane1D;49C; has the additional structure of being an object of the center ℒ(%plane1D;49C;-Coalg) of the category of coalgebras. We prove that braided groups which are reconstructed from braided monoidal categories over %plane1D;49C; are commutative algebras in the center of %plane1D;49C;-Coagl. We give further information about Hopf algebras in ℒ(%plane1D;49C;-Coalg).