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

Abstract

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.

18 pages, relation to Quillen's adjunction clarified, Theorem 5.3 changed into Proposition 5.3, Remark 5.5 extended, references added

References in corpus (6)

• Integrating L-infinity algebras
• $Q$-manifolds and Higher Analogs of Lie Algebroids
• Principal infinity-bundles - Presentations
• L_infinity algebras as 1-jets of simplicial manifolds (and a bit beyond)
• Differential gorms, differential worms
• 3-Crossed Modules of Commutative Algebras

Cited by in corpus (6)

• Semistrict Higher Gauge Theory
• Towards an M5-Brane Model II: Metric String Structures
• Higher Gauge Theory
• On the homotopy theory for Lie $\infty$-groupoids, with an application to integrating $L_{\infty }$-algebras
• Rota-type operators on null-filiform associative algebras
• Differentiating $L_{\infty }$ groupoids