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Pre-Lie deformation theory

Vladimir Dotsenko, Sergey Shadrin, Bruno Vallette

Abstract

In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the $dd^c$-lemma.

Pre-Lie deformation theory

Abstract

In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the -lemma.

Paper Structure

This paper contains 14 sections, 25 theorems, 151 equations.

Table of Contents

  1. Deformation theory with Lie algebras
  2. Deformation theory with associative algebras
  3. Example : homotopy modules over a Koszul algebra
  4. Example
  5. Deformation theory with pre-Lie algebras
  6. Remark
  7. Example : homotopy algebras over a Koszul operad
  8. Homotopy properties of algebras over an operad
  9. Categorical operadic interpretation
  10. Homotopy transfer theorem
  11. Homotopy trivialization
  12. Homotopy trivialization
  13. Homotopy transfer theorem
  14. Combinatorics of graphs

Key Result

Proposition 1

When the infinitesimal action of the Lie algebra $g_0$ defined in Section subsec:InfAction converges at $t=1$, it is given by the action of the gauge group $\Gamma$ on the variety of Maurer--Cartan elements under the formula:

Theorems & Definitions (56)

  • Proposition 1: GoldmanMillson88
  • proof
  • Proposition 2
  • proof
  • Theorem \oldthetheorem
  • proof
  • Lemma 1
  • proof
  • Proposition 3
  • proof
  • ...and 46 more