From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets

Let gg be a simplicial Lie algebra with Moore complex NgNg of length kk. Let GG be the simplicial Lie group integrating gg, such that each GnGn is simply connected. We use the 1-jet of the classifying space W¯G to construct, starting from gg, a Lie kk-algebra LL. The so constructed Lie kk-algebra LL is actually a differential graded Lie algebra. The differential and the brackets are explicitly described in terms (of a part) of the corresponding kk-hypercrossed complex structure of NgNg. The result can be seen as a geometric interpretation of Quillen’s (purely algebraic) construction of the adjunction between simplicial Lie algebras and dg-Lie algebras.