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Polydifferential Lie bialgebras and graph complexes

Vincent Wolff

Abstract

We study the deformation complex of a canonical morphism $i$ from the properad of (degree shifted) Lie bialgebras $\mathbf{Lieb}_{c,d}$ to its polydifferential version $\mathcal{D}(\mathbf{Lieb}_{c,d})$ and show that it is quasi-isomorphic to the oriented graph complex $\mathbf{GC}^{\text{or}}_{c+d+1}$, up to one rescaling class. As the latter complex is quasi-isomorphic to the original graph complex $\mathbf{GC}_{c+d}$, we conclude that the space of homotopy non-trivial infinitesimal deformations of the canonical map $i$ can be identified with the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$; moreover, every such an infinitesimal deformation extends to a genuine deformation of the canonical morphism $i$ from $\mathbf{Lieb}_{c,d}$ to $\mathcal{D}(\mathbf{Lieb}_{c,d})$. The full deformation complex is described with the help of a new graph complex of so called entangled graphs, whose suitable quotient complex is shown to contain the tensor product $H(\mathbf{GC}_c) \otimes H(\mathbf{GC}_d)$ of cohomologies of Kontsevich graph complexes $\mathbf{GC}_c \otimes \mathbf{GC}_d$.

Polydifferential Lie bialgebras and graph complexes

Abstract

We study the deformation complex of a canonical morphism from the properad of (degree shifted) Lie bialgebras to its polydifferential version and show that it is quasi-isomorphic to the oriented graph complex , up to one rescaling class. As the latter complex is quasi-isomorphic to the original graph complex , we conclude that the space of homotopy non-trivial infinitesimal deformations of the canonical map can be identified with the Grothendieck-Teichmüller Lie algebra ; moreover, every such an infinitesimal deformation extends to a genuine deformation of the canonical morphism from to . The full deformation complex is described with the help of a new graph complex of so called entangled graphs, whose suitable quotient complex is shown to contain the tensor product of cohomologies of Kontsevich graph complexes .
Paper Structure (7 sections, 6 theorems, 49 equations)

This paper contains 7 sections, 6 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Lie bialgebras
  3. Reminder on graph complexes and deformation theory of Lie bialgebras
  4. Polydifferential functor on props
  5. Properad of polydifferential Lie bialgebras
  6. Complex of entangled graphs
  7. Main theorem

Key Result

Lemma 5.2

There is a morphism of properads given by

Theorems & Definitions (18)

  • Example 5.1
  • Lemma 5.2
  • proof
  • Definition 6.1
  • Example 6.2
  • Lemma 6.3
  • proof
  • Remark 6.4
  • Theorem 6.5
  • proof
  • ...and 8 more