Bisections and cocycles on Hopf algebroids (2305.12465v3)

Abstract: We introduce left and right groups of bisections of a Hopf algebroid and show that they form a group crossed homomorphism with the group $Aut(\mathcal{L})$ of bialgebroid automorphisms. We also introduce a nonAbelian cohomology $H^2(\mathcal{L},B)$ governing cotwisting of a Hopf algebroid with base $B$. We also introduce a notion of coquasi-bialgebroid $\mathcal{L}$ via a 3-cocycle on $\mathcal{L}$. We also give dual versions of these constructions. For the Ehresmann-Schauenburg Hopf algebroid $\mathcal{L}(P,H)$ of a quantum principal bundle or Hopf-Galois extension, we show that the group of bisections reduces to the group $Aut_H(P)$ of bundle automorphisms, and give a description of the nonAbelian cohomology in concrete terms in two cases: $P$ subject to a braided' commutativity condition and $P$ a cleft extension ortrivial' bundle. Next we show that the action bialgebroid $B# H^{op}$ associated to a braided-commutative algebra $B$ in the category of $H$-crossed (or Drinfeld-Yetter) modules over a Hopf algebra $H$ is an fact a Hopf algebroid. We show that the bisection groups are again isomorphic and can be described concretely as a natural space $Z^1_{\triangleleft}(H,B)$ of multiplicative cocycles. We also do the same for the nonAbelian cohomology and for $Aut(\mathcal{L})$. We give specific results for the Heisenberg double or Weyl Hopf algebroid $H^*# H^{op}$ of $H$. We show that if $H$ is coquasitriangular then its transmutation braided group $B=\underline{H}$ provides a canonical action Hopf algebroid $\underline H# H^{op}$ and we show that if $H$ is factorisable then $\underline H# H^{op}$ is isomorphic to the Weyl Hopf algebroid of $H$. We also give constructions for coquasi versions of $\mathcal{L}(P,H)$ and of the Connes-Moscovici bialgebroid. Examples of the latter are given from the data of a subgroup $G\subseteq X$ of a finite group and choice of transversal.