Domenico Fiorenza And

We show that the mapping cone of a morphism of differential graded Lie algebras χ : L → M can be canonically endowed with an L∞-algebra structure which at the same time lifts the Lie algebra structure on L and the usual differential on the mapping cone. Moreover, this structure is unique up to isomorphisms of L∞-algebras. The associated deformation functor coincides with the one introduced by the second author in [19]. Introduction There are several cases where the tangent and obstruction spaces of a deformation theory are the cohomology groups of the mapping cone of a morphism χ : L → M of differential graded Lie algebras. It is therefore natural to ask if there exists a canonical differential graded Lie algebra structure on the complex (Cχ, δ), where Cχ = ⊕C i χ, C i χ = L i ⊕M , δ(l,m) = (dl, χ(l)− dm), such that the projection Cχ → L is a morphism of differential graded Lie algebras. In general we cannot expect the existence of a Lie structure: in fact the canonical bracket l1 ⊗ l2 7→ [l1, l2]; m⊗ l 7→ 1 2 [m,χ(l)]; m1 ⊗m2 7→ 0 satisfies the Leibniz rule with respect to the differential δ but not the Jacobi identity. However, the Jacobi identity for this bracket holds up to homotopy, and so we can look for the weaker request of a canonical L∞ structure on Cχ. More precisely, let K be a fixed characteristic zero base field, denote by DG the category of differential graded vector spaces, by DGLA the category of differential graded Lie algebras, by L∞ the category of L∞ algebras and by DGLA 2 the category of morphisms in DGLA. The four functors DGLA → L∞ Quillen construction, L∞ → DG forgetting higher brakets, DGLA → DG {L χ −→M} 7→ Cχ, DGLA → DGLA L 7→ {L → 0}, give a commutative diagram DGLA // L∞ DGLA // DG Our first result is Date: April 3, 2007. 1991 Mathematics Subject Classification. 17B70, 13D10.