On neutral Tannakian subcategories of loop quiver representations

We explored the categories of (twisted) representations of a loop quiver. These representation categories have two choices of tensor structures: Kronecker tensor and Simpson tensor. By studying the rigidity properties, we have provided several examples of (semi-) Tannakian categories using the category of (twisted) representations of a loop quiver for both tensors.

We have introduced the concept of essentially finite loop quiver bundles based on the work of Nori, Borne, and Vistoli. As an application, we have given some examples of (semi-) Tannakian categories of equivariant bundles and Hitchin pairs. Additionally, we have defined the notion of H-nflat twisted loop quiver bundles and have established Tannakian category structures for certain classes of varieties.