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automorphism infinity-Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

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Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

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Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Rational homotopy theory

… under construction

Contents

Idea

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

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

Definition

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

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

[ϕ,λ]:=ϕλλϕ:AA [\phi,\lambda] := \phi \circ \lambda - \lambda \circ \phi : A \to A

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

d Aϕ:=[d A,ϕ]:AA d_A \phi := [d_A, \phi] : A \to A

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

Definition

(automorphism \infty-Lie algebra)

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

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

  • in degree 00 the derivations that commute with the differential d Ad_A

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

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

Properties

Automorphism group

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

(…)

(See chapter 6 of Sullivan).

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

Let XX be a rational space whose Sullivan model is 𝔤\mathfrak{g}, Xexp(𝔤)X \simeq \exp(\mathfrak{g}). Let aut(𝔤)aut(𝔤)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)G(X) be the maximal reductive group of genuine automorphisms of CE(𝔤)CE(\mathfrak{g}) (see above).

Then the rational space

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

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

Examples

References

The general definition of aut(𝔤)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)Aut(A) of a dg-algebra is discussed in paragraph 6 there. Few details on proofs are given there. Only recently in

a detailed proof is given.

Concrete computations of aut(𝔤)aut(\mathfrak{g}) for some classes of rational spaces X=exp(𝔤)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

Revised on March 25, 2013 16:41:33 by Urs Schreiber (89.204.138.37)