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Cohomology of Lie coalgebras

Joseph Chuang, Andrey Lazarev, Yunhe Sheng, Rong Tang

Abstract

A Koszul duality-type correspondence between coderived categories of conilpotent differential graded Lie coalgebras and their Chevalley-Eilenberg differential graded algebras is established. This gives an interpretation of Lie coalgebra cohomology as a certain kind of derived functor. A similar correspondence is proved for coderived categories of commutative cofibrant differential graded algebras and their Harrison differential graded Lie coalgebras.

Cohomology of Lie coalgebras

Abstract

A Koszul duality-type correspondence between coderived categories of conilpotent differential graded Lie coalgebras and their Chevalley-Eilenberg differential graded algebras is established. This gives an interpretation of Lie coalgebra cohomology as a certain kind of derived functor. A similar correspondence is proved for coderived categories of commutative cofibrant differential graded algebras and their Harrison differential graded Lie coalgebras.

Paper Structure

This paper contains 17 sections, 27 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Lie coalgebras, their comodules and universal enveloping coalgebras
  4. Universal enveloping coalgebras of Lie coalgebras
  5. Conilpotent Lie coalgebras
  6. PBW theorem
  7. Koszul duality I: conilpotent Lie coalgebras
  8. Bar-cobar adjunctions
  9. Acyclic abelian Lie coalgebras
  10. Koszul complex for symmetric coalgebras
  11. Koszul duality for locally finite comodules
  12. Comodules over non-negatively graded Lie coalgebras
  13. Conilpotent Koszul duality
  14. Koszul duality II: commutative cofibrant algebras
  15. Bar-cobar adjunctions
  16. ...and 2 more sections

Key Result

Theorem 3.1

The functor $C\mapsto L(C)$ from coassociative coalgebras to Lie coalgebras admits a right adjoint, $\mathop{\mathrm{\mathfrak g}}\nolimits\mapsto \mathop{\mathrm{\operatorname{U^c}}}\limits(\mathop{\mathrm{\mathfrak g}}\nolimits)$, called the universal enveloping coalgebra of the Lie coalgebra $\ma

Theorems & Definitions (75)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • Example 3.7
  • Example 3.8
  • ...and 65 more