Affine group dg-schemes and linear representations I - Basic theory and Tannakian reconstructions

TL;DR

This paper develops the foundational theory of affine group dg-schemes, their Lie algebras, and linear representations, establishing Tannakian reconstruction theorems that recover these structures from their representation categories.

Contribution

It introduces the basic theory of affine group dg-schemes and proves Tannaka type reconstruction theorems linking these schemes to their representation categories.

Findings

Reconstruction of affine group dg-schemes from their linear representations

Development of the theory of Lie algebras associated with dg-schemes

Establishment of Tannakian duality in the dg setting

Abstract

We develop a basic theory of affine group dg-schemes, their Lie algebraic counterparts and linear representations. We prove Tannaka type reconstruction theorems that an affine group dg-scheme can be recovered from the dg-tensor category of its linear representations as well as from the rigid dg-tensor category of its finite dimensional linear representations along with the forgetful functors to the underlying dg-tensor category of cochain complexes.