Differential graded algebras over some reductive group

TL;DR

This paper investigates the properties of commutative differential graded algebras within the representation category of a reductive algebraic group, focusing on derived categories, t-structures, and Tannakian fundamental groups.

Contribution

It provides a detailed analysis of the derived category of differential graded modules and criteria for t-structures, advancing understanding in algebraic and representation theory.

Findings

Describes the derived category of differential graded modules over such algebras.

Provides a criterion for the existence of a t-structure on the derived category.

Characterizes the coordinate ring of the Tannakian fundamental group of the heart.

Abstract

We study the general properties of commutative differential graded algebras in the category of representations over a reductive algebraic group with an injective central cocharacter. Besides describing the derived category of differential graded modules over such an algebra, we also provide a criterion for the existence of a t-structure on the derived category together with a characterization of the coordinate ring of the Tannakian fundamental group of its heart.