Displaying similar documents to “The center of a graded connected Lie algebra is a nice ideal”

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...

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...

Characteristic zero loop space homology for certain two-cones

Calin Popescu (1999)

Commentationes Mathematicae Universitatis Carolinae

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Given a principal ideal domain R of characteristic zero, containing 1/2, and a two-cone X of appropriate connectedness and dimension, we present a sufficient algebraic condition, in terms of Adams-Hilton models, for the Hopf algebra F H ( Ω X ; R ) to be isomorphic with the universal enveloping algebra of some R -free graded Lie algebra; as usual, F stands for free part, H for homology, and Ω for the Moore loop space functor.

Homotopy Lie algebras and fundamental groups via deformation theory

Martin Markl, Stefan Papadima (1992)

Annales de l'institut Fourier

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We formulate first results of our larger project based on first fixing some easily accessible invariants of topological spaces (typically the cup product structure in low dimensions) and then studying the variations of more complex invariants such as π * Ω S (the homotopy Lie algebra) or gr * π 1 S (the graded Lie algebra associated to the lower central series of the fundamental group). We prove basic rigidity results and give also an application in low-dimensional topology.