Let X be a geometrically irreducible smooth projective curve defined over a field k. Assume that X has a k–rational point; fix a k–rational point x ∈ X . From these data we construct an affine group scheme GX defined over the field k as well as a principal GX–bundle EGX over the curve X . The group scheme GX is given by a Q–graded neutral Tannakian category built out of all strongly semistable ...

In this paper we study the classifying theory of principal bundles in the parametrized setting, motivated by recent interest in higher gauge theory. Using simplicial techniques, we construct a product-preserving classifying space functor for groups in the category of spaces over a fixed space B. Additionally, we prove that the fiberwise geometric realization functor sends a large class of simpl...

Given a complex manifold M equipped with a holomorphic action of a connected complex Lie group G, and a holomorphic principal H–bundle EH over X equipped with a G–connection h, we investigate the connections on the principal H–bundle EH that are (strongly) adapted to h. Examples are provided by holomorphic principal H– bundles equipped with a flat partial connection over a foliated manifold.

The Schrödinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schrödinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle – the Schrödinger line bundle. This line bundle has trivializations indexed by inertial observers and is associa...

Budzyński and Kondracki ([3]) have introduced a notion of locally trivial quantum principal fibre bundle making use of an algebraic notion of covering, which allows a reconstruction of the bundle from local pieces. Following this approach, we construct covariant differential algebras and connections on locally trivial quantum principal fibre bundles by gluing together such locally given geometr...

It is shown that every quantum principal bundle with a compact structure group is a Hopf-Galois extension. This property naturally extends to the level of general differential structures, so that every differential calculus over a quantum principal bundle with a compact structure group is a graded-differential variant of the Hopf-Galois extension.

Let G be a connected reductive group. The late Ramanathan gave a notion of (semi)stable principal G-bundle on a Riemann surface and constructed a projective moduli space of such objects. We generalize Ramanathan’s notion and construction to higher dimension, allowing also objects which we call semistable principal G-sheaves, in order to obtain a projective moduli space: a principal G-sheaf on a...

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k–rational point x0. Let ̟(M,x0) denote the corresponding fundamental group–scheme introduced by Nori. Let EG be a principal G–bundle over M , where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M . We prove that the following thr...

Let P be a parabolic subgroup of a semisimple simply connected linear algebraic group G over C and ρ an irreducible homomorphism from P to a complex reductive group H . We show that the associated principal H –bundle over G/P , associated for ρ to the principal P –bundle defined by the quotient map G −→ G/P , is stable. We describe the Harder–Narasimhan reduction of the G–bundle over G/P obtain...