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automorphism infinity-Lie algebra

… under construction …

Idea

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

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

Definition

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

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

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

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

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

Definition

(automorphism $\mathrm{\infty }$-Lie algebra)

The ∞-Lie algebra $\mathrm{aut}(A)$ is the dg-Lie algebra which

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

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

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

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

Properties

Automorphism group

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

(…)

(See chapter 6 of Sullivan).

Classifying space for $\mathrm{Aut}(X)$-principal bundles

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

Then the rational space

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

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

Examples

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

Related concepts

• deformation theory

• tangent complex

• cotangent complex

References

The general definition of $\mathrm{aut}(𝔤)$ 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 $\mathrm{Aut}(A)$ 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 $\mathrm{aut}(𝔤)$ for some classes of rational spaces $X=\exp (𝔤)$ 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