Cohomology and Extensions of $C_p$-Green Functors of Lie Type

TL;DR

This paper develops a cohomology theory for $C_p$-Green functors of Lie type, unifying algebraic and topological structures under cyclic group actions, and extends classical Lie algebra extension concepts to an equivariant context.

Contribution

It introduces a new framework combining Green functors with Lie algebra structures under cyclic group actions, including cohomology and extension theories.

Findings

Defines equivariant Chevalley-Eilenberg cohomology for Green functors

Establishes a correspondence between extensions and second cohomology groups

Generalizes classical Lie algebra extension theory to the equivariant setting

Abstract

We develop a theory of $C_p$-Green functors of Lie type, unifying the axiomatic framework of Green functors with the structure of Lie algebras under the action of a cyclic group $C_p$ of prime order. Extending classical notions from representation theory and topology, we define tensor and exterior products, introduce an equivariant Chevalley-Eilenberg cohomology, and construct cup products that endow the cohomology with a graded Green functor of Lie type structure. A key result establishes a correspondence between equivalence classes of singular extensions and second cohomology groups, generalizing classical Lie algebra extension theory to the equivariant setting. This framework enriches the toolkit for studying equivariant algebraic structures and paves the way for further applications in deformation theory, homotopical algebra, and representation theory.