Bisections and cocycles on Hopf algebroids

Abstract

We introduce and study the group $\mathcal{B}(\mathcal{L})$ of bisections of a Hopf algebroid $\mathcal{L}$ and show that they form a group crossed module or $2$-group with the group $\mathrm{Aut}(\mathcal{L})$ of automorphisms. Moreover, the group of vertical bisections turns out to be part of a certain non-Abelian cohomology ${\mathcal{H}}^2(\mathcal{L},B)$ governing cotwisting of a Hopf algebroid with base $B$. For the Ehresmann–Schauenburg Hopf algebroid $\mathcal{L}(P,H)$ of a quantum principal bundle or Hopf–Galois extension, $\mathcal{B}(\mathcal{L}(P,H))$ reduces to the group ${\mathrm{Aut}}_H(P)$ of bundle automorphisms and vertical bisections to the group of ‘gauge transformations’ of the bundle. The general ${\mathcal{H}}^2(\mathcal{L}(P,H),B)$ 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 $B\#H^{\mathrm{op}}$ 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 $\underline{H}\#H^{\mathrm{op}}$ when $H$ is coquasitriangular and $\underline{H}$ is its transmutation.