Hopf algebroids and twists for quantum projective spaces

Abstract

We study the relationship between antipodes on a Hopf algebroid ▪ in the sense of Böhm–Szlachanyi and the group of twists that lies inside the associated convolution algebra. We specialize to the case of a faithfully flat H-Hopf–Galois extensions B⊆A and related Ehresmann–Schauenburg bialgebroid. In particular, we find that the twists are in one-to-one correspondence with H-comodule algebra automorphism of A. We work out in detail the U(1)-extension O(CPqn−1)⊆O(Sq2n−1) on the quantum projective space and show how to get an antipode on the bialgebroid out of the K-theory of the base algebra O(CPqn−1).