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Homotopies d’algèbres de Lie et de leurs algèbres enveloppantes. (Homotopies of Lie algebras and their envelopping algebras). (French) Zbl 0682.55008
Algebraic topology, rational homotopy, Proc. Conf., Louvain-la- Neuve/Belg. 1986, Lect. Notes Math. 1318, 26-30 (1988).
[For the entire collection see Zbl 0652.00011.]
The author has proved the following theorem: Two morphisms f and g in DGL are homotopic in DGL if and only if Uf and Ug are homotopic in DGA. (Here, DGL denotes the category of connected graded differentiable Lie algebras of finite type (Q-algebras) as defind by Quillen. $$U\Lambda$$ denotes the envelopping algebra of a graded differentiable Lie algebra $$\Lambda$$.) The above theorem is motivated by the construction of Adams- Hilton which associates to certain CW-complexes a differentiable graded associative $${\mathbb{Z}}$$-algebra which reflects in a way the structure of the CW-complex.
Reviewer: S.-N.Patnaik

##### MSC:
 55P62 Rational homotopy theory 55P35 Loop spaces 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18G55 Nonabelian homotopical algebra (MSC2010)