Let and be differential graded Lie algebras and be a morphism of graded vector spaces. Define as
for any . The morphism is called a Cartan homotopy if it satisfies the two conditions
This name has an evident geometric origin: if is the tangent sheaf of a smooth manifold and is the sheaf of complexes of differential forms, then the contraction of differential forms with vector fields is a Cartan homotopy
In this case, is the Lie derivative along the vector field , and the conditions and , together with the defining equation and with the equations and expressing the fact that is a dgla morphism, are nothing but the well-known Cartan identities involving contractions and Lie derivatives.
It is a straightforward computation to see that, if is a Cartan homotopy, then the degree zero morphism of graded vector spaces is actually a dgla morphism.
D. Fiorenza, M. Manetti. L-∞ algebras, Cartan homotopies and period maps; arXiv:math/0605297
D. Fiorenza, M. Manetti. A period map for generalized deformations. Journal of Noncommutative Geometry, Vol. 3, No. 4 (2009), 579-597; arXiv:0808.0140.
D. Fiorenza, E. Martinengo. A short note on ∞-groupoids and the period map for projective manifolds Publications of the nLab. Vol. 2 (2012); arXiv:0911.3845
Created on September 12, 2012 at 23:58:50. See the history of this page for a list of all contributions to it.