homotopy hypothesis-theorem
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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 is
a pair of Lie algebras and
equipped with two Lie algebra homomorphisms
(to the Lie algebra of Lie derivations)
such that for all we have
.
Notice that the Lie algebra structure on 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 of . This leads over to the following perspective.
Equivalently, a differential crossed module is a dg-Lie algebra structure on a chain complex concentrated in degrees 0 and 1.
The components of the dg-Lie bracket are
the given bracket ;
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.
The notion of Lie algebra crossed modules is due to:
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 September 4, 2024 at 16:43:00. See the history of this page for a list of all contributions to it.