Braided bi-Galois theory II: The cocommutative case

International audienceWe study groups of bi-Galois objects over a Hopf algebra H ina braided monoidal category B. We assume H to be cocommutativein a certain sense; this does not mean that H is a cocommutativecoalgebra with respect to the braiding given in B , but it iscocommutative with respect to a different braiding subject tospecific axioms. The type of cocommutative Hopf algebras underconsideration (investigated in previous papers) occurs naturally, forexample in Majid’s transmutation construction. We show that forcocommutative H the suitably defined cocommutative bi-Galoisobjects form a subgroup in the group of H-H-bi-Galois objects. Wealso show that all cocycles on H are lazy, and that second (lazy)cohomology describes the subgroup of cleft bi-Galois extensions inthe group of cocommutative ones