nLab
inner derivation Lie 2-algebra

Idea

For inner derivation Lie 2-algebra $\mathrm{inn}(𝔤)$ of a Lie algebra $𝔤$ is the (strict) Lie 2-algebra equivalently given as

• the dg-Lie algebra of contractions ${\iota }_x$ and inner derivations ${ℒ}_x=[d_{\mathrm{CE}},x]$ acting on the Chevalley-Eilenberg algebra of $𝔤$;

• the differential crossed module $(𝔤\stackrel{\mathrm{Id}}{\to }𝔤)$ with the action being the adjoint action of $𝔤$ on itself;

• the dual of the Weil algebra $W(𝔤)$.

In the first formulation this may be identified with the dg-Lie algebra whose

• elements in degree -1 are the contractions ${\iota }_x:\mathrm{CE}(𝔤)\to \mathrm{CE}(𝔤)$ with $x\in 𝔤$;

• elements in degree 0 are the inner derivations ${ℒ}_x=[d_{\mathrm{CE}(𝔤)},{\iota }_x]:\mathrm{CE}(𝔤)\to \mathrm{CE}(𝔤)$;

• the differential $\partial :𝔤\to 𝔤$ is given by the commutator $\partial =[d_{\mathrm{CE}(𝔤)},-]$;

• the bracket is the graded commutator bracket of derivations:

  • $[{\iota }_x,{\iota }_y]=0$

  • $[{ℒ}_x,{\iota }_y]={\iota }_{[x,y]}$

  • $[{ℒ}_x,{ℒ}_y]={ℒ}_{[x,y]}$.

So this is the full subalgebra of the automorphism ∞-Lie algebra of $\mathrm{CE}(𝔤)$ on the inner derivations.

See Weil algebra as CE-algebra of inner derivations for more details.

Properties

• The first formulation makes manifest that $\mathrm{inn}(𝔤)$ is the structure that has historically been called Cartan calculus.

• In the (∞,1)-category of ∞-Lie algebras $\mathrm{inn}(𝔤)$ is equivalent to the point. See Weil algebra for details on this.

References

The structure of $\mathrm{inn}(𝔤)$ 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, $L_{\mathrm{\infty }}$ algebra connections and applications to String- and Chern-Simons $n$-transport (ref)