nLab differential crossed module

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 $\mathfrak{g}$ is

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

• equipped with two Lie algebra homomorphisms

  • $\partial : \mathfrak{g}_1 \to \mathfrak{g}_0$

  • $\rho : \mathfrak{g}_0 \to Der(\mathfrak{g}_1)$

    (to the Lie algebra of Lie derivations)

• such that for all $x \in \mathfrak{g}_0, b,b' \in \mathfrak{g}_1$ we have

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

  • $\rho(\partial b)(b') = [b, b']$.

Notice that the Lie algebra structure on $\mathfrak{g}_1$ 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 $\mathfrak{g}_1$ of $\mathfrak{g}_0$. 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 $(\mathfrak{g}_1 \stackrel{\partial}{\to} \mathfrak{g}_0)$ concentrated in degrees 0 and 1.

The components of the dg-Lie bracket are

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

• the mixed bracket

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

  identifies with the action:

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

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

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

is equivalently the above condition

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

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.

Related concepts

• Lie algebra, Lie group

• Lie 2-algebra, Lie 2-group

• L-infinity algebra, smooth ∞-group

• Lie algebroid, Lie groupoid

• L-∞ algebroid, smooth ∞-groupoid

References

See also the references at Lie 2-algebra.

The notion of Lie algebra crossed modules is due to:

• Christian Kassel, Jean-Louis Loday: appendix of: Extensions centrales d’algèbres de Lie, Annales de l’Institut Fourier, 32 4 (1982) 119-142 [numdam:AIF_1982__32_4_119_0]

On the history of this and related concepts:

• Johannes Huebschmann, On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras, J. Geom. Mech. 13 (2021), 385-402 [arXiv:2208.02539]

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.

• George Janelidze, Internal crossed modules, Georgian Math. J. 10 (2003) 99114 (journal)

Further discussion:

• Friedrich Wagemann: On Lie algebra crossed modules, Comm. Algebra 34 5 (2006) 1699-1722 [arXiv:math/0611375, doi:10.1080/00927870500542705]