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Graph complexes and Deformation theories of the (wheeled) properads of quasi- and pseudo-Lie bialgebras

Oskar Frost

Abstract

Quasi-Lie bialgebras are natural extensions of Lie-bialgebras, where the cobracket satisfies the co-Jacobi relation up to some natural obstruction controlled by a skew-symmetric 3-tensor $φ$. This structure was introduced by Drinfeld while studying deformation theory of universal enveloping algebras and has since seen many other applications in algebra and geometry. In this paper we study the derivation complex of strongly homotopy quasi-Lie bialgebra, both in the unwheeled (i.e standard) and wheeled case, and compute its cohomology in terms of Kontsevich graph complexes.

Graph complexes and Deformation theories of the (wheeled) properads of quasi- and pseudo-Lie bialgebras

Abstract

Quasi-Lie bialgebras are natural extensions of Lie-bialgebras, where the cobracket satisfies the co-Jacobi relation up to some natural obstruction controlled by a skew-symmetric 3-tensor . This structure was introduced by Drinfeld while studying deformation theory of universal enveloping algebras and has since seen many other applications in algebra and geometry. In this paper we study the derivation complex of strongly homotopy quasi-Lie bialgebra, both in the unwheeled (i.e standard) and wheeled case, and compute its cohomology in terms of Kontsevich graph complexes.

Paper Structure

This paper contains 23 sections, 45 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction and outline
  2. Introduction
  3. Outline
  4. Notation
  5. Lie bialgebras and deformation complexes of properads
  6. Pseudo- and quasi-Lie bialgebras
  7. Properad extensions
  8. Wheelification
  9. Derivation complexes
  10. Graph complexes
  11. The directed Kontsevich's graph complex
  12. Subcomplexes of $\mathsf{dGC}_k$
  13. The bi-weighted graph complex
  14. Subcomplexes of $\mathsf{fwGC}_k$, $\mathsf{fwQGC}_k$ and $\mathsf{fwPGC}_k$
  15. Loop number zero
  16. ...and 8 more sections

Key Result

Theorem \oldthetheorem

For any $c,d\in\mathbb{Z}$, there is an explicit morphism of dg Lie algebras which is a quasi-isomorphism up to a rescaling class.

Figures (2)

  • Figure 1: Example of recursive removal of special-in vertices of the rightmost graph. All vertices except the top one are special in-vertices.
  • Figure 2: Example of a graph in $S_1^{in}\mathsf{owGC}_k$ where out-core vertices are colored gray and the special out-vertices are colored white.

Theorems & Definitions (101)

  • Theorem \oldthetheorem: Main Theorem I
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem: Main Theorem II
  • Theorem \oldthetheorem: Main Theorem III
  • Theorem \oldthetheorem: Main Theorem IV
  • Theorem \oldthetheorem: Main Theorem V
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 91 more