Hopf-Galois extensions and twisted Hopf algebroids

Abstract

We propose a new notion of antipode S for a left Hopf algebroid which does not assume antimultiplicativity. We also show that if L is a left Hopf algebroid then so is its cotwist Lς as an extension of a previous bialgebroid Drinfeld cotwist theory. We show that the Ehresmann-Schauenburg bialgebroid of a quantum principal bundle P or Hopf-Galois extension with structure quantum group H is in fact a left Hopf algebroid L(P,H) and that it has an antipode S at least if H is coquasitriangular or if the bundle is trivial so that P=B#σH is a cleft extension and the cocycle-action (▹,σ) is of associative type. We also show in the latter case that L(B#σH,H)=L(B#H,H)σ˜ for a Hopf algebroid cotwist ς=σ˜. Thus, introducing a σ of associative type appears at the Hopf algebroid level as a Drinfeld cotwist. We view the affine quantum group Uq(sl2)ˆ and the quantum Weyl group of uq(sl2) as examples of associative type.