Crossed modules, quantum braided groups and ribbon structures

Previous results about crossed modules over a braided Hopf algebra are applied for to study of quantum groups in braided categories. Cross products for braided Hopf algebras and quantum braided groups are built. Criteria when a braided Hopf algebra or a quantum group is a cross product are obtained. A generalization of the Majid's trunsmutation procedure for quantum braided groups is considered. A ribbon structure on a quantum braided group and its compatibility with cross product and transmutation are studied.