From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets
2012
Abstract
Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, such that each G(n) is simply connected.
We use the 1-jet of the classifying space (W) over barG 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. (C) 2012 Elsevier B.V.
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