differential crossed module


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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.


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.


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.

Revised on August 28, 2011 13:05:51 by Urs Schreiber (