nLab differential crossed module

Context

Higher category theory

higher category theory

1-categorical presentations

Homological algebra

homological algebra

Introduction

diagram chasing

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 $\mathfrak{g}$ 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.

References

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 (89.204.153.64)