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Integration theory of Lie-graph algebras

Ricardo Campos, Bruno Vallette

TL;DR

The paper addresses integrating deformation problems controlled by differential graded Lie algebras arising from properadic structures by introducing Lie-graph algebras—a Lie-type framework encoded by directed graphs that acts on the totalization of a properad. It develops an explicit integration theory, including a graph-based exponential map, a deformation gauge group, and a concrete formula for the gauge action on Maurer-Cartan elements, thereby describing the Deligne groupoid in this broader setting. Key contributions include proving an isomorphism between the gauge group and the deformation gauge group via the graph exponential, providing a closed graphical formula for MC action, and clarifying that the Lie-graph operad is not finitely generated, connecting to pre-Lie and rooted-tree structures. The framework generalizes operadic cases and furnishes tools for deformation theory of gebras over properads, with further applications to Deligne groupoids and twisting in a sequel.

Abstract

We develop the Lie theory of Lie-admissible algebras whose product is enriched with higher operations modeled on directed graphs with a view to apply it to the deformation theories controlled by this kind of Lie algebras. We produce effective formulas for their exponential map, their gauge group structure and the action on Maurer--Cartan elements. The main motivation and range of applications lies in the deformation theory of types of bialgebras which is done in a sequel article. This work extends the case of pre-Lie algebra structures which appear in the deformation theory of operadic algebras.

Integration theory of Lie-graph algebras

TL;DR

The paper addresses integrating deformation problems controlled by differential graded Lie algebras arising from properadic structures by introducing Lie-graph algebras—a Lie-type framework encoded by directed graphs that acts on the totalization of a properad. It develops an explicit integration theory, including a graph-based exponential map, a deformation gauge group, and a concrete formula for the gauge action on Maurer-Cartan elements, thereby describing the Deligne groupoid in this broader setting. Key contributions include proving an isomorphism between the gauge group and the deformation gauge group via the graph exponential, providing a closed graphical formula for MC action, and clarifying that the Lie-graph operad is not finitely generated, connecting to pre-Lie and rooted-tree structures. The framework generalizes operadic cases and furnishes tools for deformation theory of gebras over properads, with further applications to Deligne groupoids and twisting in a sequel.

Abstract

We develop the Lie theory of Lie-admissible algebras whose product is enriched with higher operations modeled on directed graphs with a view to apply it to the deformation theories controlled by this kind of Lie algebras. We produce effective formulas for their exponential map, their gauge group structure and the action on Maurer--Cartan elements. The main motivation and range of applications lies in the deformation theory of types of bialgebras which is done in a sequel article. This work extends the case of pre-Lie algebra structures which appear in the deformation theory of operadic algebras.

Paper Structure

This paper contains 10 sections, 26 theorems, 157 equations, 16 figures.

Table of Contents

  1. Algebraic structures on convolution algebras
  2. Properadic calculus
  3. Lie-graph algebras
  4. Lie-graph algebra structure on properadic convolution algebra
  5. The complete setting
  6. Integration theory of Lie-graph algebras
  7. Deformation gauge group
  8. Graph exponential map
  9. Action on Maurer--Cartan elements
  10. Proof of Proposition \ref{['prop:nofinitepresentation']}

Key Result

Lemma 1.3

The infinitesimal composition map of a dg properad $(\mathcal{P}, \mathrm{d}, \gamma, \eta)$ induces a dg Lie-admissible algebra structure $\left( \oplus_{m,n\in \mathbb{N}} \mathcal{P}(m,n)^{\mathbb{S}_m^{\mathrm{op}}\times \mathbb{S}_n}, \mathrm{d}, \star \right)$ on the total sum of the inv

Figures (16)

  • Figure 1: Examples of 2-leveled connected graph directed from top to bottom.
  • Figure 2: The first directed simple graphs.
  • Figure 3: Example of a partial composition in the operad $\mathrm{Lie}\textrm{-}\mathrm{gra}$.
  • Figure 4: Example of a partial composition in the operad $\mathrm{RT}$.
  • Figure 5: Example of a partial composition of ladders.
  • ...and 11 more figures

Theorems & Definitions (72)

  • Definition 1.1: Properad and coproperad
  • Lemma 1.3: MerkulovVallette09I
  • Definition 1.4: Infinitesimal coproperad
  • Proposition 1.5: Convolution algebra MerkulovVallette09I
  • Definition 1.6: Convolution monoid
  • Definition 1.7: Maurer--Cartan element
  • Proposition 1.8: MerkulovVallette09I
  • Definition 1.10: Directed simple graph
  • Lemma 1.11
  • proof
  • ...and 62 more