nLab
inner derivation Lie 2-algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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

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

  • elements in degree -1 are the contractions ι x:CE(𝔤)CE(𝔤)\iota_x : CE(\mathfrak{g}) \to CE(\mathfrak{g}) with x𝔤x \in \mathfrak{g};

  • elements in degree 0 are the inner derivations x=[d CE(𝔤),ι x]:CE(𝔤)CE(𝔤)\mathcal{L}_x = [d_{CE(\mathfrak{g})}, \iota_x] : CE(\mathfrak{g}) \to CE(\mathfrak{g});

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

  • the bracket is the graded commutator bracket of derivations:

    • [ι x,ι y]=0[\iota_x, \iota_y] = 0

    • [ x,ι y]=ι [x,y][\mathcal{L}_x, \iota_y] = \iota_{[x,y]}

    • [ x, y]= [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(𝔤)CE(\mathfrak{g}) on the inner derivations.

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

Properties

References

The structure of inn(𝔤)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

Revised on September 20, 2010 17:04:41 by Urs Schreiber (188.20.66.18)