The notion of quantum group equivariant homogeneous vector bundles on quantum homogeneous spaces is introduced. The category of such quantum vector bundles is shown to be exact, and its Grothendieck group is determined. It is also shown that the algebras of functions on quantum homogeneous spaces are noetherian.

Let X be a smooth projective curve over C (you can think of X as a Riemann surface in the analytic setting if you wish) throughout. Then there exists a smooth variety PicX over C whose C-points are the isomorphism classes of line bundles on X. More generally, for any scheme (variety) S over C, the set HomC(S,PicX) corresponds to a sheafification of the line bundles on the family of curves X ×S ...

The problem of constructing moduli space of vector bundles over a projective manifold has attracted many mathematicians for decades. In mid 60’s Mumford first constructed the moduli space of vector bundles over algebraic curves via his celebrated GIT machinery. Later, in early 80’s Atiyah and Bott found an infinite dimensional symplectic quotient description of this moduli space. Since then, we...

Let X be an elliptic curve over an algebraically closed field. We prove that some exact sub-categories of the category of vector bundles over X, defined using Harder-Narasimhan filtrations, have the same K-groups as the whole category.

The notion of extension of a given C*-category C by a C*-algebra A is introduced. In the commutative case A = C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging to the initial category. It is shown that the Doplicher-Roberts algebra (DR-algebra in the following) associated to an object in the extension of a strict tensor C*-category is a con...

The stratified vector bundles on a smooth variety defined over an algebraically closed field k form a neutral Tannakian category over k. We investigate the affine group–scheme corresponding to this neutral Tannakian category.

We give a method to construct stable vector bundles whose rank divides the degree over curves of genus bigger than one. The method complements the one given by Newstead. Finally, we make some systematic remarks and observations in connection with rationality of moduli spaces of stable vector bundles.