nLab differential crossed module

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Contents

Idea

The notion of differential crossed module (or crossed module of/in Lie algebras) is a way to encode the structure of a strict Lie 2-algebra in terms of two ordinary Lie algebras.

This is the infinitesimal version of how a smooth crossed module encodes a smooth strict 2-group.

Definition

As crossed modules of Lie algebras

A differential crossed module 𝔤\mathfrak{g} is

  • a pair of Lie algebras 𝔤 0\mathfrak{g}_0 and 𝔤 1\mathfrak{g}_1

  • equipped with two Lie algebra homomorphisms

    • :𝔤 1𝔤 0\partial : \mathfrak{g}_1 \to \mathfrak{g}_0

    • ρ:𝔤 0Der(𝔤 1) \rho : \mathfrak{g}_0 \to Der(\mathfrak{g}_1)

      (to the Lie algebra of Lie derivations)

  • such that for all x𝔤 0,b,b𝔤 1x \in \mathfrak{g}_0, b,b' \in \mathfrak{g}_1 we have

    • (ρ(x)(b))=[x,(b)]\partial ( \rho(x)(b) ) = [x, \partial(b)]

    • ρ(b)(b)=[b,b]\rho(\partial b)(b') = [b, b'].

Notice that the Lie algebra structure on 𝔤 1\mathfrak{g}_1 is already fixed by the rest of the data. So a differential crossed module may equivalently be thought of as extra structure on a Lie module 𝔤 1\mathfrak{g}_1 of 𝔤 0\mathfrak{g}_0. This leads over to the following perspective.

As dg-Lie algebras

Equivalently, a differential crossed module is a dg-Lie algebra structure on a chain complex (𝔤 1𝔤 0)(\mathfrak{g}_1 \stackrel{\partial}{\to} \mathfrak{g}_0) concentrated in degrees 0 and 1.

The components of the dg-Lie bracket are

  • the given bracket [,]:𝔤 0𝔤 0𝔤 0[-,-] : \mathfrak{g}_0 \otimes \mathfrak{g}_0 \to \mathfrak{g}_0;

  • the mixed bracket

    [,]:𝔤 0𝔤 1𝔤 1 [-,-] : \mathfrak{g}_0 \otimes \mathfrak{g}_1 \to \mathfrak{g}_1

    identifies with the action:

    [x,b]:=ρ(x)(b). [x,b] := \rho(x)(b) \,.

This way the respect of the dg-bracket for the differential

[x,b]=[x,b]+[x,b]=[x,b] \partial [x,b] = [\partial x, b ] + [x, \partial b] = [x,\partial b]

is equivalently the above condition

ρ(x)(b)=ρ(x)(b). \partial \rho(x)(b) = \rho(x)(\partial b) \,.

As strict Lie 2-algebras

By the discussion there, dg-Lie algebras are strict L-∞ algebras (those for which all the brackets of higher arity vanish). Therefore the above identification of differential crossed modules with 2-term dg-Lie algebras identifies these also with strict Lie 2-algebras.

References

See also the references at Lie 2-algebra.

On the history of this and related concepts:

The crossed modules (in groups or Lie groups) and the differential crossed modules are examples of the internal crossed modules. A good theory of them is developed in semiabelian categories.

Last revised on August 4, 2023 at 09:59:07. See the history of this page for a list of all contributions to it.