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strict Lie 2-algebra

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∞-Lie theory

∞-Lie theory

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Lie theory

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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) of an ordinary Lie algebra 𝔤. This is the one coming from the differential crossed module (𝔤→adder(𝔤)).

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