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Integrating L-infinity algebras

Andre Henriques

TL;DR

The paper develops a comprehensive framework to integrate L_infty-algebras into Lie n-groups by constructing Kan simplicial manifolds ∫L from Chevalley–Eilenberg data, and it analyzes the resulting simplicial homotopy. It shows that ∫L is Kan and encodes higher homotopy via H_{n-1}(L), extends the construction to Banach-manifold settings to manage infinite-dimensional cases, and introduces Postnikov-type truncations that yield Lie n-groups under concrete discreteness criteria. The string Lie 2-algebra str(g) is treated in depth, producing a model whose truncation τ_≤2 ∫str has the homotopy type of B String, and giving explicit, bundle-theoretic descriptions of its simplices. Overall, the work provides a robust integration procedure for L_infty-algebras, connects to existing string-group models, and clarifies the hierarchy between Lie 2-groups and higher coherent structures.

Abstract

Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group. In the case of the string Lie 2-algebra of Baez and Crans, this recovers the model of the string group introduced in math.QA/0504123 .

Integrating L-infinity algebras

TL;DR

The paper develops a comprehensive framework to integrate L_infty-algebras into Lie n-groups by constructing Kan simplicial manifolds ∫L from Chevalley–Eilenberg data, and it analyzes the resulting simplicial homotopy. It shows that ∫L is Kan and encodes higher homotopy via H_{n-1}(L), extends the construction to Banach-manifold settings to manage infinite-dimensional cases, and introduces Postnikov-type truncations that yield Lie n-groups under concrete discreteness criteria. The string Lie 2-algebra str(g) is treated in depth, producing a model whose truncation τ_≤2 ∫str has the homotopy type of B String, and giving explicit, bundle-theoretic descriptions of its simplices. Overall, the work provides a robust integration procedure for L_infty-algebras, connects to existing string-group models, and clarifies the hierarchy between Lie 2-groups and higher coherent structures.

Abstract

Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group. In the case of the string Lie 2-algebra of Baez and Crans, this recovers the model of the string group introduced in math.QA/0504123 .

Paper Structure

This paper contains 14 sections, 34 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Homotopy Lie algebras
  3. The string group and its Lie algebra
  4. Homotopy Lie groups
  5. Integrating $L_\infty$-algebras
  6. Acknowledgments
  7. Kan simplicial objects
  8. Simplicial homotopy groups
  9. Banach manifolds
  10. Integrating $L_\infty$-algebras
  11. The homotopy groups of ${ \int} L$
  12. Integrating Lie $n$-algebras
  13. The string Lie 2-algebra
  14. Appendix

Key Result

Lemma 2.4

Let $S\subset\Delta[n]$ be a collapsable simplicial set. Let $f:X\to Y$ be a map between reduced simplicial objects that satisfies the conditions of Definition dmk for all $m<n$ (this implicitly means that $Hom(\Lambda[m,j]\to\Delta[m],X\to Y)$ is representable). Then the object of commutative squar

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.4
  • Definition 2.1
  • Definition 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 38 more