Realization of Lie algebras and classifying spaces of crossed modules

Félix, Yves; Tanré, Daniel

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.48550/arXiv.2103.04927

Title :

Realization of Lie algebras and classifying spaces of crossed modules

Author(s) :

Félix, Yves [Auteur]
Université Catholique de Louvain = Catholic University of Louvain [UCL]
Tanré, Daniel [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Algebraic and Geometric Topology

Publisher :

Mathematical Sciences Publishers

Publication date :

2024

ISSN :

1472-2747

English keyword(s) :

Algebraic Topology (math.AT)
FOS: Mathematics
55P62
17B55
55U10

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of non-simply connected spaces. In particular, there is a realization functor, $\langle -\rangle$, ...
Show more >The category of complete differential graded Lie algebras provides nice algebraic models for the rational homotopy types of non-simply connected spaces. In particular, there is a realization functor, $\langle -\rangle$, of any complete differential graded Lie algebra as a simplicial set. In a previous article, we considered the particular case of a complete graded Lie algebra, $L_{0}$, concentrated in degree 0 and proved that $\langle L_{0}\rangle$ is isomorphic to the usual bar construction on the Malcev group associated to $L_{0}$. Here we consider the case of a complete differential graded Lie algebra, $L=L_{0}\oplus L_{1}$, concentrated in degrees 0 and 1. We establish that the category of such two-stage Lie algebras is equivalent to explicit subcategories of crossed modules and Lie algebra crossed modules, extending the equivalence between pronilpotent Lie algebras and Malcev groups. In particular, there is a crossed module $\mathcal{C}(L)$ associated to $L$. We prove that $\mathcal{C}(L)$ is isomorphic to the Whitehead crossed module associated to the simplicial pair $(\langle L\rangle, \langle L_{0}\rangle)$. Our main result is the identification of $\langle L\rangle$ with the classifying space of $\mathcal{C}(L)$.Show less >

Language :

Anglais

Popular science :

Non

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

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