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

TL;DR

This paper develops a categorical framework to study algebraic structures over fields, defining a field of invariants and constructing generic forms, thereby generalizing previous results for algebras and Hopf algebras.

Contribution

It introduces a symmetric monoidal category approach to classify forms of algebraic structures and constructs generic forms over a base algebra, extending existing theories.

Findings

Defined the field of invariants for algebraic structures

Constructed generic forms over a base algebra

Applied framework to classify two-cocycles on Hopf algebras

Abstract

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 $\mathcal{C}_W$, which gives rise to a subfield $K_0\subseteq K$, 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 $\mathcal{C}_W$ is a $K_0$-form of $Rep_{\bar{K}}(Aut(W))$, and we use it to construct a generic form $\widetilde{W}$ over a commutative $K_0$ algebra $B_W$ (so that forms of $W$ are exactly the specializations of $\widetilde{W}$). 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 can one use the construction to classify two-cocycles on some finite dimensional Hopf algebras.