Displaying similar documents to “Characteristic zero loop space homology for certain two-cones”

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 ) 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 ) 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...

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 of rational numbers. An ideal I in the Lie algebra H ( 𝕃 ( V ) , d ) is called if, for every cycle α 𝕃 ( V ) such that [ α ] belongs to I , the kernel of the map H ( 𝕃 ( V ) , d ) H ( 𝕃 ( V 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 x ) , d ) . We apply this computation for the determination of the rational homotopy Lie algebra L X = π * ( Ω X ) of a simply connected space X . We deduce...

Knit products of graded Lie algebras and groups

Michor, Peter W.

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Let A = k A k and B = k B k be graded Lie algebras whose grading is in 𝒵 or 𝒵 2 , but only one of them. Suppose that ( α , β ) is a derivatively knitted pair of representations for ( A , B ) , i.e. α and β satisfy equations which look “derivatively knitted"; then A B : = k , l ( A k B l ) , endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra A ( α , β ) B . This graded Lie algebra is called the knit product of A and B . The author investigates the general situation for any graded Lie subalgebras A and...

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 ) 𝑘 induces a morphism of algebras I : H H * ( 𝒞 * ( M ) ; 𝒞 * ( M ) ) 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...