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

Preprint English OPEN
Jurco, Branislav (2011)
  • Related identifiers: doi: 10.1016/j.geomphys.2012.09.002
  • Subject: Mathematical Physics | Mathematics - Differential Geometry
    arxiv: Mathematics::Algebraic Topology | Mathematics::Category Theory

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial 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 is actually a differential graded Lie algebra. The differential and the brackets are explicitly described in terms (of a part) of the corresponding k-hypercrossed complex structure of Ng. 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.
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