Effective integration of Lie type algebras

TL;DR

This survey articulates an effective program for integrating Lie-type algebras beyond the classical Lie setting. It develops a unifying framework where direct exponential-type maps, gauge actions, and explicit graph- and operad-based products yield concrete, computable integration formulas for $\mathrm{pre-}\mathrm{L}$ie, $\mathrm{Lie}$-graph, $\mathrm{L}_\infty$, and absolute curved $\mathrm{EL}_\infty$-algebras. The approach extends to higher homotopies via $\mathrm{sL}_\infty$-algebras, with a canonical integration functor into $\infty$-groupoids, including a universal Maurer–Cartan algebra and higher Baker–Campbell–Hausdorff structures, and pushes into prime characteristic using $\mathrm{E}$-coalgebras and curved cooperads for $p$-adic homotopy. Overall, the work provides algorithmic, explicit models for deformation problems and morphisms in higher Lie-type settings, offering tools for rational and $p$-adic homotopy theory and beyond.

Abstract

This is a short survey on the recent developments made in the integration theory with effective formulas of algebraic structures stronger or higher than Lie algebras.

Figures (7)

• Figure 1: The first directed simple graphs.
• Figure 2: Example of a partial composition in the operad $\mathrm{Lie}\textrm{-}\mathrm{graph}$.
• Figure 3: The first terms of the graph circle product $\circledcirc$ .
• Figure 4: The first terms of the graph exponential and logarithm maps.
• Figure 5: The first terms of the bowtie element $(\mathbb{1}+x)\stackrel{\alpha}{\Join} (\mathbb{1}+y)$ .

Theorems & Definitions (42)

• Definition 1.1: Complete differential graded Lie algebra
• Definition 1.2: Maurer--Cartan element
• Definition 1.3: Gauge equivalence
• Theorem 1.4: Baker--Campbell-Hausdorff, see BF12
• Definition 1.5: Gauge group
• Proposition 1.6: Gauge group action
• proof : Sketch of proof
• Definition 2.1: Pre-Lie algebra
• Definition 2.2: Circle product
• Theorem 2.3: DotsenkoShadrinVallette16