∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
… under construction …
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.
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
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
is a derivation of degree $-(k+1)$.
(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 $[\phi,\lambda] = \phi \circ \lambda - \lambda \circ \phi$.
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).
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
is the classifying space for $Aut(X)$-principal bundles, i.e. for bundles with typical fiber $X$.
The general definition of $aut(\mathfrak{g})$ is the topic of p. 313 (45 of 63) and following in
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
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
Last revised on March 7, 2017 at 09:50:59. See the history of this page for a list of all contributions to it.