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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.
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.
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
identifies with the action:
This way the respect of the dg-bracket for the differential
is equivalently the above condition
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.
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.