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Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Dietrich Burde (2006)

Open Mathematics

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs...

Leibniz A -algebras

David A. Towers (2020)

Communications in Mathematics

A finite-dimensional Lie algebra is called an A -algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras.

Leibniz cohomology for differentiable manifolds

Jerry M. Lodder (1998)

Annales de l'institut Fourier

We propose a definition of Leibniz cohomology, H L * , for differentiable manifolds. Then H L * becomes a non-commutative version of Gelfand-Fuks cohomology. The calculations of H L * ( R n ; R ) reduce to those of formal vector fields, and can be identified with certain invariants of foliations.

Les ( a , b ) -algèbres à homotopie près

Walid Aloulou (2010)

Annales mathématiques Blaise Pascal

On étudie dans cet article les notions d’algèbre à homotopie près pour une structure définie par deux opérations . et [ , ] . Ayant déterminé la structure des G ∞ algèbres et des P ∞ algèbres, on généralise cette construction et on définit la stucture des ( a , b ) -algèbres à homotopie près. Etant donnée une structure d’algèbre commutative et de Lie différentielle graduée pour deux décalages des degrés donnés par a et b , on donnera une construction explicite de l’algèbre à homotopie près associée et on précisera...

Lie algebraic characterization of manifolds

Janusz Grabowski, Norbert Poncin (2004)

Open Mathematics

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.

Lie algebras

George Seligman (1987)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

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