Differential Calculi on Quantum Principal Bundles over Projective Bases

Abstract: We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of covariant bimodules together with a morphism of sheaves -- the differential -- such that the Leibniz rule and surjectivity hold locally. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. In a second step we introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. In this case Hopf--Galois extensions of algebras lift to Hopf--Galois extensions of exterior algebras with compatible differentials. In particular, the examples of principal (covariant) calculi on the quantum principal bundles $\mathcal{O}_q(\mathrm{SL}_2(\mathbb{C}))$ and $\mathcal{O}_q(\mathrm{GL}_2(\mathbb{C}))$ over the projective space $\mathrm{P}^1(\mathbb{C})$ are discussed in detail.