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A note on Poisson derivations

Jiantao Li (2018)

Czechoslovak Mathematical Journal

Free Poisson algebras are very closely connected with polynomial algebras, and the Poisson brackets are used to solve many problems in affine algebraic geometry. In this note, we study Poisson derivations on the symplectic Poisson algebra, and give a connection between the Jacobian conjecture with derivations on the symplectic Poisson algebra.

Associative and Lie deformations of Poisson algebras

Elisabeth Remm (2012)

Communications in Mathematics

Considering a Poisson algebra as a nonassociative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this nonassociative algebra. We give a natural interpretation of deformations which preserve the underlying associative structure and of deformations which preserve the underlying Lie algebra and we compare the associated cohomologies with the Poisson cohomology parametrizing the general deformations of Poisson algebras.

Binary operations in classical and quantum mechanics

Janusz Grabowski, Giuseppe Marmo (2003)

Banach Center Publications

Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply properties which are usually additionally required.

Divergence operators and odd Poisson brackets

Yvette Kosmann-Schwarzbach, Juan Monterde (2002)

Annales de l’institut Fourier

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, Δ , of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples...

Enveloping algebras of Slodowy slices and the Joseph ideal

Alexander Premet (2007)

Journal of the European Mathematical Society

Let G be a simple algebraic group over an algebraically closed field 𝕜 of characteristic 0, and 𝔤 = Lie G . Let ( e , h , f ) be an 𝔰 𝔩 2 -triple in 𝔤 with e being a long root vector in 𝔤 . Let ( · , · ) be the G -invariant bilinear form on 𝔤 with ( e , f ) = 1 and let χ 𝔤 * be such that χ ( x ) = ( e , x ) for all x 𝔤 . Let 𝒮 be the Slodowy slice at e through the adjoint orbit of e and let H be the enveloping algebra of 𝒮 ; see [31]. In this article we give an explicit presentation of H by generators and relations. As a consequence we deduce that H contains an ideal...

Growth of some varieties of Leibniz-Poisson algebras

Ratseev, S. M. (2011)

Serdica Mathematical Journal

2010 Mathematics Subject Classification: 17A32, 17B63.Let V be a variety of Leibniz-Poisson algebras over an arbitrary field whose ideal of identities contains the identities {{x1,y1},{x2,y2},ј,{xm,ym}} = 0, {x1,y1}·{x2,y2}· ј ·{xm,ym} = 0 for some m. It is shown that the exponent of V exists and is an integer.

Homologie et modèle minimal des algèbres de Gerstenhaber

Grégory Ginot (2004)

Annales mathématiques Blaise Pascal

On étudie ici les notions d’algèbre de Gerstenhaber à homotopie près et d’homologie des algèbres de Gerstenhaber du point de vue de la théorie des opérades. Précisément, on donne une description explicite des 𝒢 -algèbres à homotopie près (c’est-à-dire d’algèbres sur le modèle minimal de l’opérade 𝒢 des algèbres de Gerstenhaber). On décrit également le complexe calculant l’homologie des 𝒢 -algèbres. On donne une suite spectrale qui converge vers cette homologie et quelques exemples de calculs. Enfin...

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.

Isomorphisms of Poisson and Jacobi brackets

Janusz Grabowski (2000)

Banach Center Publications

We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as, for example, those given by Poisson or contact structures. We admit degenerate structures as well, which seems to be new in the literature.

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...

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...

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