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Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G bethe simplicial Lie group integrating g, which is simply connected in eachsimplicial level. We use the 1-jet of the classifying space of G to construct,starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L isactually a differential graded Lie algebra. The differential and the bracketsare explicitly described in terms (of a part) of the correspondingk-hypercrossed complex structure of Ng. The result can be seen as a geometricinterpretation of Quillen's (purely algebraic) construction of the adjunctionbetween simplicial Lie algebras and dg-Lie algebras.

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