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 generalized characters. We specialize to the case of a faithfully flat Hopf-Galois extensions B ⊆ A and associated Ehresmann-Schauenburg bialgebroid. In particular we find that the twists are in one-to-one correspondence with unital comodule algebra endomorphisms of A. We work out in details the extension A(CP n−1 q ) ⊆ A(S 2n−1 q ) on the quantum projective space showing how to construct an antipode on the bialgebroid starting form the K-theory of the base algebra A(CP n−1 q ).