Homotopy Lie algebras and fundamental groups via deformation theory

Martin Markl; Stefan Papadima

Displaying similar documents to “Homotopy Lie algebras and fundamental groups via deformation theory”

On the rational homotopy Lie algebra of spaces with finite dimensional rational cohomology and homotopy

Martin Markl (1989)

Annales de l'institut Fourier

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The problem of the characterization of graded Lie algebras which admit a realization as the homotopy Lie algebra of a space of type $F$ is discussed. The central results are formulated in terms of varieties of structure constants, several criterions for concrete algebras are also deduced.

Homotopy Lie algebras; lower central series and the Koszul property.

Papadima, Stefan, Suciu, Alexander I. (2004)

Geometry & Topology

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The Hochschild cohomology of a closed manifold

Yves Felix, Jean-Claude Thomas, Micheline Vigué-Poirrier (2004)

Publications Mathématiques de l'IHÉS

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Let M be a closed orientable manifold of dimension and $𝒞^{*} (M)$ be the usual cochain algebra on M with coefficients in a field. The Hochschild cohomology of M, $H H^{*} (𝒞^{*} (M); 𝒞^{*} (M))$ is a graded commutative and associative algebra. The augmentation map $ε : 𝒞^{*} (M) \to 𝑘$ induces a morphism of algebras $I : H H^{*} (𝒞^{*} (M); 𝒞^{*} (M)) \to H H^{*} (𝒞^{*} (M); 𝑘)$ . In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H H^{*} (𝒞^{*} (M); 𝑘)$ , which is in general quite small. The algebra $H H^{*} (𝒞^{*} (M); 𝒞^{*} (M))$ is expected to...

The center of a graded connected Lie algebra is a nice ideal

Yves Félix (1996)

Annales de l'institut Fourier

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Let $(𝕃 (V), d)$ be a free graded connected differential Lie algebra over the field $ℚ$ of rational numbers. An ideal $I$ in the Lie algebra $H (𝕃 (V), d)$ is called if, for every cycle $α \in 𝕃 (V)$ such that $[α]$ belongs to $I$ , the kernel of the map $H (𝕃 (V), d) \to H (𝕃 (V \oplus ℚ x), d)$ , $d (x) = α$ , is contained in $I$ . We show that the center of $H (𝕃 (V), d)$ is a nice ideal and we give in that case some informations on the structure of the Lie algebra $H (𝕃 (V \oplus ℚ x), d)$ . We apply this computation for the determination of the rational homotopy Lie algebra $L_{X} = π_{*} (Ω X) \otimes ℚ$ of a simply connected space $X$ . We deduce...

Graded Lie Algebras of depth one.

Rikard Bogvad, Carl Jacobsson (1990)

Manuscripta mathematica

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On commutation relations for 3-graded Lie algebras.

de Oliveira, M.P. (2001)

The New York Journal of Mathematics [electronic only]

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On the homology of free Lie algebras

Calin Popescu (1998)

Commentationes Mathematicae Universitatis Carolinae

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Given a principal ideal domain $R$ of characteristic zero, containing $1 / 2$ , and a connected differential non-negatively graded free finite type $R$ -module $V$ , we prove that the natural arrow $𝕃 F H (V) \to F H 𝕃 (V)$ is an isomorphism of graded Lie algebras over $R$ , and deduce thereby that the natural arrow $U F H 𝕃 (V) \to F H U 𝕃 (V)$ is an isomorphism of graded cocommutative Hopf algebras over $R$ ; as usual, $F$ stands for free part, $H$ for homology, $𝕃$ for free Lie algebra, and $U$ for universal enveloping algebra. Related facts and examples are also...