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From Poisson algebras to Gerstenhaber algebras

Yvette Kosmann-Schwarzbach (1996)

Annales de l'institut Fourier

Constructing an even Poisson algebra from a Gerstenhaber algebra by means of an odd derivation of square 0 is shown to be possible in the category of Loday algebras (algebras with a non-skew-symmetric bracket, generalizing the Lie algebras, heretofore called Leibniz algebras in the literature). Such “derived brackets” give rise to Lie brackets on certain quotient spaces, and also on certain Abelian subalgebras. The construction of these derived brackets explains the origin of the Lie bracket on...

Geometry of third order ODE systems

Alexandr Medvedev (2010)

Archivum Mathematicum

We compute cohomology spaces of Lie algebras that describe differential invariants of third order ordinary differential equations. We prove that the algebra of all differential invariants is generated by 2 tensorial invariants of order 2, one invariant of order 3 and one invariant of order 4. The main computational tool is a Serre-Hochschild spectral sequence and the representation theory of semisimple Lie algebras. We compute differential invariants up to degree 2 as application.

Gerstenhaber and Batalin-Vilkovisky algebras; algebraic, geometric, and physical aspects

Claude Roger (2009)

Archivum Mathematicum

We shall give a survey of classical examples, together with algebraic methods to deal with those structures: graded algebra, cohomologies, cohomology operations. The corresponding geometric structures will be described(e.g., Lie algebroids), with particular emphasis on supergeometry, odd supersymplectic structures and their classification. Finally, we shall explain how BV-structures appear in Quantum Field Theory, as a version of functional integral quantization.

Homology and modular classes of Lie algebroids

Janusz Grabowski, Giuseppe Marmo, Peter W. Michor (2006)

Annales de l’institut Fourier

For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evens, J.-H. Lu, and A. Weinstein.

Le complexe de Koszul en algèbre et topologie

Stephen Halperin (1987)

Annales de l'institut Fourier

The Koszul complex, as introduced in 1950, was a differential graded algebra which modelled a principal fibre bundle. Since then it has been an effective tool, both in algebra and in topology, for the calculation of homological and homotopical invariants. After a partial summary of these results we recall more recent generalizations of this complex, and some applications.

Left-right noncommutative Poisson algebras

José Casas, Tamar Datuashvili, Manuel Ladra (2014)

Open Mathematics

The notions of left-right noncommutative Poisson algebra (NPlr-algebra) and left-right algebra with bracket AWBlr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NPlr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWBlr and NPlr-algebras the notions of actions, representations, centers, actors and crossed modules are described as...

Currently displaying 61 – 80 of 146