∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
For inner derivation Lie 2-algebra $inn(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ is the (strict) Lie 2-algebra equivalently given as
the dg-Lie algebra of contractions $\iota_x$ and inner derivations $\mathcal{L}_x = [d_{CE},x]$ acting on the Chevalley-Eilenberg algebra of $\mathfrak{g}$;
the differential crossed module $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ with the action being the adjoint action of $\mathfrak{g}$ on itself;
the dual of the Weil algebra $W(\mathfrak{g})$.
In the first formulation this may be identified with the dg-Lie algebra whose
elements in degree -1 are the contractions $\iota_x : CE(\mathfrak{g}) \to CE(\mathfrak{g})$ with $x \in \mathfrak{g}$;
elements in degree 0 are the inner derivations $\mathcal{L}_x = [d_{CE(\mathfrak{g})}, \iota_x] : CE(\mathfrak{g}) \to CE(\mathfrak{g})$;
the differential $\partial : \mathfrak{g} \to \mathfrak{g}$ is given by the commutator $\partial = [d_{CE(\mathfrak{g})}, -]$;
the bracket is the graded commutator bracket of derivations:
$[\iota_x, \iota_y] = 0$
$[\mathcal{L}_x, \iota_y] = \iota_{[x,y]}$
$[\mathcal{L}_x, \mathcal{L}_y] = \mathcal{L}_{[x,y]}$.
So this is the full subalgebra of the automorphism ∞-Lie algebra of $CE(\mathfrak{g})$ on the inner derivations.
See Weil algebra as CE-algebra of inner derivations for more details.
The first formulation makes manifest that $inn(\mathfrak{g})$ is the structure that has historically been called Cartan calculus.
In the (∞,1)-category of ∞-Lie algebras $inn(\mathfrak{g})$ is equivalent to the point. See Weil algebra for details on this.
The structure of $inn(\mathfrak{g})$ is of course in itself very simple and goes as such back at least to Cartan.
Its role as a Lie 2-algebra in the context of ∞-Chern-Weil theory has been discussed in section 6 of
Last revised on September 20, 2010 at 17:04:41. See the history of this page for a list of all contributions to it.