nLab automorphism infinity-Lie algebra

… under construction …

Idea

The automorphism $\infty$-Lie algebra $aut(\mathfrak{g})$ of an ∞-Lie algebra $\mathfrak{g}$ – or dually $aut(CE(\mathfrak{g}))$ of the corresponding Chevalley-Eilenberg algebra – has in degree $k$ the derivations on $CE(\mathfrak{g})$ of degree $-k$. The higher Lie algebra version of the automorphism Lie algebra of an ordinary Lie algebra.

In terms of rational homotopy theory $aut(\mathfrak{g})$ is a model for the rationalization of the group of automorphismss of the rational space $\exp(\mathfrak{g})$ corresponding to $CE(\mathfrak{g})$ under the Sullivan construction.

Definition

Let $A := (\wedge^\bullet \mathfrak{a}^*, d_A)$ be a semifree dg-algebra of finite type.

Notice that for $\phi : A \to A$ a derivation of degree $-k$ and $\lambda : A\to A$ another derivation of degree $-l$ the commutator

$$[\phi,\lambda] := \phi \circ \lambda - \lambda \circ \phi : A \to A$$

is itself a derivation, of degree $-(k+l)$. In particular, since the differential $d_A : A \to A$ is itself a derivation of degree +1, we have that

$$d_A \phi := [d_A, \phi] : A \to A$$

is a derivation of degree $-(k+1)$.

Definition

(automorphism $\infty$-Lie algebra)

The ∞-Lie algebra $aut(A)$ is the dg-Lie algebra which

• in degree $-k$ for $k \gt 0$ has the derivations $\phi : A \to A$ of degree $-k$;

• in degree $0$ the derivations that commute with the differential $d_A$

• whose differential $\delta_{aut(A)} := [d_A,-]$ is given by the commutator with the differential of $A$;

• whose Lie bracket is the commutator $[\phi,\lambda] = \phi \circ \lambda - \lambda \circ \phi$.

Properties

Automorphism group

For stating the fundamental theorem about $aut(\mathfrak{g})$ below we need some facts about the ordinary automorphism group of a dg-algebra $A$.

(…)

(See chapter 6 of Sullivan).

Classifying space for $Aut(X)$-principal bundles

Let $X$ be a rational space whose Sullivan model is $\mathfrak{g}$, $X \simeq \exp(\mathfrak{g})$. Let $aut'(\mathfrak{g}) \subset aut(\mathfrak{g})$ be the sub dg-algebra of the automorphism $\infty$-Lie algebra on the maximal nilpotent ideal in degree 0. Let $G(X)$ be the maximal reductive group of genuine automorphisms of $CE(\mathfrak{g})$ (see above).

Then the rational space

$$\exp(aut'(\mathfrak{g}))/G(X) \simeq B Aut (X)$$

is the classifying space for $Aut(X)$-principal bundles, i.e. for bundles with typical fiber $X$.

Examples

• The inner derivation Lie 2-algebra $inn(\mathfrak{g})$ is the full subalgebra of the automorphism $\infty$-Lie algebra on the inner derivations of the Chevalley-Eilenberg algebra of a Lie algebra $\mathfrak{g}$.

Related concepts

• automorphism Lie algebra

• deformation theory

• tangent complex

• cotangent complex

References

The general definition of $aut(\mathfrak{g})$ is the topic of p. 313 (45 of 63) and following in

• Dennis Sullivan, Infinitesimal computations in topology Publications Mathématiques de l’IHÉS, 47 (1977) (numdam)

The automorphism group $Aut(A)$ of a dg-algebra is discussed in paragraph 6 there. Few details on proofs are given there. Only recently in

• Andrey Lazarev, Jonathan Block, …

a detailed proof is given.

Concrete computations of $aut(\mathfrak{g})$ for some classes of rational spaces $X = \exp(\mathfrak{g})$ can be found for instance in

• Samual Bruce Smith, The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces , Journal of Pure and Applied Algebra Volume 160, Issues 2-3, 25 June 2001, Pages 333-343