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
is itself a derivation, of degree . In particular, since the differential is itself a derivation of degree +1, we have that
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
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