On weak Lie 2-algebras

TL;DR

This paper explores the structure of weak Lie 2-algebras, their relation to 2-term homotopy Lie algebras, and their applications as infinitesimal symmetries in various geometric contexts.

Contribution

It extends the theory of Lie 2-algebras to weak cases, establishing equivalences with homotopy Lie algebras and connecting to geometric structures like Poisson and Courant algebroids.

Findings

Equivalence between Lie 2-algebras and 2-term homotopy Lie algebras.

Weak Lie 2-algebras generalize strict cases and relate to $L_ olinebreak_ olinebreak ext{infinity}$-algebras.

Applications to symmetries in differential graded Lie algebras and geometric structures.

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.