Hopf algebroids and twists for quantum projective spaces

Abstract

We study the relationship between antipodes on a Hopf algebroid $\mathcal{H}$ 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 $B\subseteq A$ 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 $\mathcal{O}(\mathbb{C}{P}_q^{n-1})\subseteq \mathcal{O}(S_q^{2n-1})$ on the quantum projective space and show how to get an antipode on the bialgebroid out of the $K$-theory of the base algebra $\mathcal{O}(\mathbb{C}{P}_q^{n-1})$.

References in corpus (6)

• Hopf algebras, cyclic cohomology and the transverse index theory
• Strong Connections on Quantum Principal Bundles
• Projective module description of the q-monopole
• Translation map in quantum principal bundles
• Anti-selfdual Connections On The Quantum Projective Plane: Instantons
• Hopf algebroids associated to Jacobi algebras

Cited by in corpus (1)

• Grothendieck-Verdier functors