Displaying similar documents to “Knit products of graded Lie algebras and groups”

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 ∈ 𝕃 ( V ) such that [ α ] belongs to I , the kernel of the map H ( 𝕃 ( V ) , d ) → H ( 𝕃 ( V ⊕ 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 ⊕ ℚ x ) , d ) . We apply this computation for the determination of the rational homotopy Lie algebra L X = π * ( Ω X ) ⊗ ℚ of a simply connected space X . We deduce...

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...

Isomorphisms between graded Frobenius algebras constructed from twisted superpotentials

Xuejun Xia, Libin Li (2022)

Czechoslovak Mathematical Journal

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In order to distinguish the connected graded Frobenius algebras determined by different twisted superpotentials, we introduce the nondegeneracy of twisted superpotentials. We give the sufficient and necessary condition for connected graded Frobenius algebras determined by two nondegenerate twisted superpotentials to be isomorphic. As an application, we classify the connected ℤ -graded Frobenius algebra of length 3, whose dimension of the degree 1 is 2.

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.