On weak Lie 2-algebras

Abstract

A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and 2-term "homotopy everything" Lie algebras; for strictly skew-symmetric Lie 2-algebras, these reduce to $L_{\infty }$-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer--Cartan equation in some differential graded Lie algebras and $L_{\infty }$-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.

Based on my talk at the Bialowieza workshop, July 2007

References in corpus (2)

• Lie theory for nilpotent L-infinity algebras
• Courant Algebroids and Strongly Homotopy Lie Algebras

Cited by in corpus (25)

• An Invitation to Higher Gauge Theory
• Teleparallel Gravity as a Higher Gauge Theory
• Higher Symplectic Geometry
• Leibniz 2-algebras and twisted Courant algebroids
• Six-Dimensional (1,0) Superconformal Models and Higher Gauge Theory
• Extended Riemannian Geometry I: Local Double Field Theory
• Courant-Dorfman algebras and their cohomology
• Lie 2-bialgebras
• The Pontryagin Class for Pre-Courant Algebroids
• Symmetric homotopy theory for operads
• Lie 2-algebra models
• Omni-Lie 2-algebras and their Dirac structures
• Topological Field Theories induced by twisted R-Poisson structure in any dimension
• Categorified Symplectic Geometry and the String Lie 2-Algebra
• L-infinity algebras and higher analogues of Dirac structures and Courant algebroids
• Courant algebroids from categorified symplectic geometry
• Higher Gauge Theory
• On the infinity category of homotopy Leibniz algebras
• Higher categorified algebras versus bounded homotopy algebras
• Deformation quantization of Leibniz algebras
• Group actions on dg-manifolds and Exact Courant algebroids
• On distributive laws in derived bracket construction and homotopy theory of derived bracket Leibniz algebras
• Derived brackets and sh Leibniz algebras
• Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids
• Higher Vector Bundles