Formal Lie groupoids
Rational homotopy theory
… under construction …
The automorphism -Lie algebra of an ∞-Lie algebra – or dually of the corresponding Chevalley-Eilenberg algebra – has in degree the derivations on of degree .
In terms of rational homotopy theory is a model for the rationalization of the group of automorphismss of the rational space corresponding to under the Sullivan construction.
Let be a semifree dg-algebra of finite type.
Notice that for a derivation of degree and another derivation of degree the commutator
[\phi,\lambda] := \phi \circ \lambda - \lambda \circ \phi : A \to A
is itself a derivation, of degree . In particular, since the differential is itself a derivation of degree +1, we have that
d_A \phi := [d_A, \phi] : A \to A
is a derivation of degree .
(automorphism -Lie algebra)
The ∞-Lie algebra is the dg-Lie algebra which
in degree for has the derivations of degree ;
in degree the derivations that commute with the differential
whose differential is given by the commutator with the differential of ;
whose Lie bracket is the commutator .
For stating the fundamental theorem about below we need some facts about the ordinary automorphism group of a dg-algebra .
(See chapter 6 of Sullivan).
Classifying space for -principal bundles
Let be a rational space whose Sullivan model is , . Let be the sub dg-algebra of the automorphism -Lie algebra on the maximal nilpotent ideal in degree 0. Let be the maximal reductive group of genuine automorphisms of (see above).
Then the rational space
B Aut (X)
is the classifying space for -principal bundles, i.e. for bundles with typical fiber .
The general definition of 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 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 for some classes of rational spaces 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