nLab inner derivation Lie 2-algebra

Idea

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.

Properties

• 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.

References

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport (ref)