Bisections and cocycles on Hopf algebroids

We introduce and study the group B(L) of bisections of a Hopf algebroid L and show that they form a group crossed module or 2 -group with the group Aut(L) of automorphisms. Moreover, the group of vertical bisections turns out to be part of a certain non-Abelian cohomology H²(L,B) governing cotwisting of a Hopf algebroid with base B . For the Ehresmann–Schauenburg Hopf algebroid L(P,H) of a quantum principal bundle or Hopf–Galois extension, B(L(P,H)) reduces to the group AutH(P) of bundle automorphisms and vertical bisections to the group of ‘gauge transformations’ of the bundle. The general H²(L(P,H),B) reduces to a known non-Abelian cohomology in the case where P is a trivial principal bundle or cleft extension. Parallel characterisations are obtained for the bisections and non-Abelian cohomology of the action Hopf algebroid B# Hᵒᵖ associated with a braided-commutative algebra B in the category of Drinfeld–Yetter modules over a Hopf algebra H . Examples include the Heisenberg double or Weyl Hopf algebroid of a Hopf algebra and a canonical action Hopf algebroid {H}# Hᵒᵖ when H is coquasitriangular and {H} is its transmutation.