Mixed graded structure on Chevalley-Eilenberg functors

TL;DR

This paper provides a formal, model-independent $$-categorical construction of the mixed graded structure on Chevalley-Eilenberg complexes, enhancing understanding of their homology and cohomology of Lie algebras.

Contribution

It introduces a purely $$-categorical, model-independent framework for the mixed graded structure on Chevalley-Eilenberg complexes, filling a gap in the literature.

Findings

Constructed Chevalley-Eilenberg $$-functors in detail

Studied formal properties of these functors

Presented conjectures on their behavior

Abstract

In this paper, we shall provide a purely $\infty$-categorical construction of the mixed graded structure over Chevalley-Eilenberg complexes computing homology and cohomology of Lie algebras defined over a field $\Bbbk$ of characteristic $0$. While this additional piece of structure on Chevalley-Eilenberg complexes is expected, and already described in terms of explicit models given by chain complexes, there is not a completely formal and model independent description of the mixed graded Chevalley-Eilenberg $\infty$-functors in available literature. After constructing in all details the Chevalley-Eilenberg $\infty$-functors and studying their main formal properties, we present some further conjectures on their behavior.