Descent, fields of invariants, and generic forms via symmetric monoidal categories

Let W be a finite dimensional algebraic structure (e.g. an algebra) over a field K of characteristic zero. We study forms of W by using Deligne's Theory of symmetric monoidal categories. We construct a category CW, which gives rise to a subfield K0⊆K, which we call the field of invariants of W. This field will be contained in any subfield of K over which W has a form. The category CW is a K0-form of RepK¯(Aut(W)), and we use it to construct a generic form W˜ over a commutative K0-algebra BW (so that forms of W are exactly the specializations of W˜). This generalizes some generic constructions for central simple algebras and for H-comodule algebras. We give some concrete examples arising from associative algebras and H-comodule algebras. As an application, we also explain how one can use the construction to classify two-cocycles on some finite dimensional Hopf algebras.