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Mixed graded structure on Chevalley-Eilenberg functors

Emanuele Pavia

TL;DR

This work provides a purely ∞-categorical construction of mixed graded structures on Chevalley–Eilenberg complexes for Lie algebras over a field of characteristic $0$, defining homological and cohomological mixed graded CE functors ${ m CE}_{oldsymbol{ m ext{ef}}}$ and ${ m CE}^{oldsymbol{ m ext{ef}}}$ whose Tate realizations recover the classical CE (co)homology. It places CE as functors into mixed graded (co)algebras and analyzes module-valued variants ${ m CE}_{oldsymbol{ m ext{ef}}}( g; -)$ and ${ m CE}^{oldsymbol{ m ext{ef}}}( g; -)$, proving functorial properties and discussing their interplay with representations and realizations. The paper further develops a framework of conjectures about fully faithful embeddings of Lie algebras and their representations into categories of mixed graded coalgebras and modules, linking to Beilinson–Bernstein–differential geometry perspectives and derived deformation theory. These constructions aim to provide model-independent, ∞-categorical tools for formal geometry and deformation problems, with potential applications to derived algebroids and D-modules on derived schemes. The results lay groundwork for a unified, functorial approach to mixed graded Chevalley–Eilenberg theory and its representations in derived settings.

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.

Mixed graded structure on Chevalley-Eilenberg functors

TL;DR

This work provides a purely ∞-categorical construction of mixed graded structures on Chevalley–Eilenberg complexes for Lie algebras over a field of characteristic , defining homological and cohomological mixed graded CE functors and whose Tate realizations recover the classical CE (co)homology. It places CE as functors into mixed graded (co)algebras and analyzes module-valued variants and , proving functorial properties and discussing their interplay with representations and realizations. The paper further develops a framework of conjectures about fully faithful embeddings of Lie algebras and their representations into categories of mixed graded coalgebras and modules, linking to Beilinson–Bernstein–differential geometry perspectives and derived deformation theory. These constructions aim to provide model-independent, ∞-categorical tools for formal geometry and deformation problems, with potential applications to derived algebroids and D-modules on derived schemes. The results lay groundwork for a unified, functorial approach to mixed graded Chevalley–Eilenberg theory and its representations in derived settings.

Abstract

In this paper, we shall provide a purely -categorical construction of the mixed graded structure over Chevalley-Eilenberg complexes computing homology and cohomology of Lie algebras defined over a field of characteristic . 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 -functors in available literature. After constructing in all details the Chevalley-Eilenberg -functors and studying their main formal properties, we present some further conjectures on their behavior.
Paper Structure (14 sections, 19 theorems, 257 equations)

This paper contains 14 sections, 19 theorems, 257 equations.

Table of Contents

  1. Mixed graded modules
  2. Main definitions and notations
  3. Algebras and coalgebras in the mixed graded setting
  4. Lie algebras in characteristic $0$
  5. Main definitions and notations
  6. Representations of Lie algebras
  7. Classical Chevalley-Eilenberg complexes
  8. Mixed graded Chevalley-Eilenberg modules
  9. The homological Chevalley-Eilenberg $\infty$-functor
  10. The cohomological Chevalley-Eilenberg $\infty$-functor
  11. Representations and mixed graded modules
  12. Conjectures and open problems
  13. Lie algebras as mixed graded coalgebras
  14. Representations as mixed graded modules

Key Result

Theorem 1

Theorems & Definitions (79)

  • Theorem : Propositions \ref{['prop:promotionCEcocommutativealgebra']}, \ref{['prop:CEcohomologicalalgebra']}, \ref{['prop:CEmod']}
  • Definition 1.1.2
  • Definition 1.1.8: CPTVV
  • Remark 1.1.13
  • Definition 1.2.2
  • Remark 1.2.16
  • Remark 1.2.18
  • Proposition 1.2.22
  • Lemma 1.2.24
  • proof
  • ...and 69 more