nLab
strict Lie 2-algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A strict Lie 2-algebra is the infinitesimal approximation to a smooth strict 2-group in generalization of how an ordinary Lie algebra is the infinitesimal approximation to a Lie group.

Definition

The notion of strict Lie 2-algebra is the special case of a general notion Lie 2-algebra for which the Jacobi identity does hold (and not just up to nontrivial isomorphism).

More precisely: a strict Lie 2-algebra is an ∞-Lie algebra with generators just in degree 1 and 2 and at most the unary and binary brackets being nontrivial.

Equivalently, this is a dg-Lie algebra with generators in the lowest two degrees.

In direct analogy to how strict 2-groups are equivalently encoded in a smooth crossed module of groups, a strict Lie 2-algebra is equivalently encoded in a differential crossed module of ordinary Lie algebras.

Examples

Derivation Lie 2-algebra

The Lie version of a smooth automorphism 2-group is the derivation Lie 2-algebra Der(𝔤)Der(\mathfrak{g}) of an ordinary Lie algebra 𝔤\mathfrak{g}. This is the one coming from the differential crossed module (𝔤adder(𝔤))(\mathfrak{g} \stackrel{ad}{\to} der(\mathfrak{g})).

Revised on August 28, 2011 12:57:51 by Urs Schreiber (89.204.153.64)