related by the Dold-Kan correspondence
dg-Lie algebras may be thought of as the “strict” strong homotopy Lie algebras. As such they support a homotopy theory. The model category structure on dg-Lie algebras is one way to present this homotopy theory. This is used for instance in deformation theory, see at formal moduli problems.
For dg-Lie algebras in positive degree and over the rational numbers this model structure, due to (Quillen 69) is one of the algebraic models for presenting rational homotopy theory (see there) of simply connected topological spaces.
is the canonical dg-Lie algebra structure on the tensor product.
dg-Lie algebras with this model structure are a rectification of L-∞ algebras: for the Lie operad and its standard cofibrant resolution, algebras over an operad over in chain complexes are dg-Lie algebras and algebras over are L-∞ algebras and by the rectification result discussed at model structure on dg-algebras over an operad there is an induced Quillen equivalence
There is also a Quillen equivalence from the model structure on dg-Lie algebras to the model structure on dg-coalgebras. This is part of a web of Quillen equivalences that identifies dg-Lie algebra/-algebras with infinitesimal derived ∞-stacks (“formal moduli problems”). More on this is at model structure for L-∞ algebras.
between dg-coalgebras and dg-Lie algebras, where the right adjoint is the (non-full) inclusion that regards a dg-Lie algebra as a differential graded coalgebra with co-binary differential, and where the left adjoint (“rectification”) sends a dg-coalgebra to a dg-Lie algebra whose underlying graded Lie algebra is the free Lie algebra on the underlying chain complex. Over a field of characteristic 0, this adjunction is a Quillen equivalence between the model structure for L-∞ algebras on and the model structure on (Hinich 98, theorem 3.2).
In particular, therefore the composite is a resolution functor for -algebras.
The model structure on dg-Lie algebras goes back to
For more discussion see
Vladimir Hinich, Homological algebra of homotopy algebras, Comm. in algebra, 25(10)(1997), 3291–3323.
and section 2.1 of