Tannakian formalism over fields with operators

TL;DR

This paper extends Tannakian formalism to tensor categories over fields with operators, including differential and difference fields, providing a unified framework for their representation theory.

Contribution

It introduces a Tannakian formalism over fields with operators, generalizing classical algebraic and differential algebraic cases.

Findings

Develops a theory of tensor categories over fields with operators.

Describes the category of representations of linear groups over such fields.

Extends classical Tannakian formalism to new settings.

Abstract

We develop a theory of tensor categories over a field endowed with abstract operators. Our notion of a "field with operators", coming from work of Moosa and Scanlon, includes the familiar cases of differential and difference fields, Hasse-Schmidt derivations, and their combinations. We develop a corresponding Tannakian formalism, describing the category of representations of linear groups defined over such fields. The paper extends the previously know (classical) algebraic and differential algebraic Tannakian formalisms.