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Formal deformations, cohomology theory and $L_\infty[1]$-structures for differential Lie algebras of arbitrary weight

Weiguo Lyu, Zihao Qi, Jian Yang, Guodong Zhou

Abstract

Generalising a previous work of Jiang and Sheng, a cohomology theory for differential Lie algebras of arbitrary weight is introduced. The underlying $L_\infty[1]$-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between $L_\infty[1]$-structures for absolute and relative differential Lie algebras are established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras. In a forthcoming paper, we will show that the operad of homotopy (relative) differential Lie algebras is the minimal model of the operad of (relative) differential Lie algebras.

Formal deformations, cohomology theory and $L_\infty[1]$-structures for differential Lie algebras of arbitrary weight

Abstract

Generalising a previous work of Jiang and Sheng, a cohomology theory for differential Lie algebras of arbitrary weight is introduced. The underlying -structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between -structures for absolute and relative differential Lie algebras are established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras. In a forthcoming paper, we will show that the operad of homotopy (relative) differential Lie algebras is the minimal model of the operad of (relative) differential Lie algebras.
Paper Structure (23 sections, 29 theorems, 134 equations)

This paper contains 23 sections, 29 theorems, 134 equations.

Table of Contents

  1. Deformation theory and minimal models
  2. Differential Lie algebras, old and new
  3. Layout of the paper
  4. Differential Lie algebras of arbitrary weight and their cohomology theory
  5. Notations
  6. Differential Lie algebras with weight and their representations
  7. Cohomology of differential operators
  8. Cohomology of differential Lie algebras
  9. Abelian extensions of differential Lie algebras
  10. Abelian extensions
  11. Classificaiton for Abelian extensions
  12. Deformations of differential Lie algebras
  13. $L_\infty[1]$-structure for (relative and absolute) differential Lie algebras
  14. $L_\infty[1]$-algebras
  15. A generalised version of derived bracket technique
  16. ...and 8 more sections

Key Result

Proposition 1.6

Let $(V,\rho, {\mathrm{d}}_V)$ be a representation of the differential Lie algebra $({\frak g}, {\mathrm{d}}_{\frak g})$. Then $({\frak g}\oplus V, {\mathrm{d}}_{\frak g}+ {\mathrm{d}}_V)$ is a differential Lie algebra, where the Lie algebra structure on ${\frak g}\oplus V$ is given by This new differential Lie algebra is denoted by ${\frak g}\ltimes V$, called the trivial extension of ${\frak g}

Theorems & Definitions (81)

  • Definition 1
  • Definition 2
  • Definition 3: CC22JS23
  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Proposition 1.6
  • Lemma 1.7
  • ...and 71 more