Left-right noncommutative Poisson algebras

José Casas; Tamar Datuashvili; Manuel Ladra

Open Mathematics (2014)

  • Volume: 12, Issue: 1, page 57-78
  • ISSN: 2391-5455

Abstract

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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 special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NPlr-algebras and AWBlr (resp. of NPr-algebras and AWBr) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWBr, the Hochschild or/and Leibniz cohomological dimension of P is ≤ n are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.

How to cite

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José Casas, Tamar Datuashvili, and Manuel Ladra. "Left-right noncommutative Poisson algebras." Open Mathematics 12.1 (2014): 57-78. <http://eudml.org/doc/269479>.

@article{JoséCasas2014,
abstract = {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 special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NPlr-algebras and AWBlr (resp. of NPr-algebras and AWBr) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWBr, the Hochschild or/and Leibniz cohomological dimension of P is ≤ n are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.},
author = {José Casas, Tamar Datuashvili, Manuel Ladra},
journal = {Open Mathematics},
keywords = {Poisson algebra; Algebras with bracket; Leibniz algebra; Representation; Left-right noncommutative Poisson algebra cohomology; Hochschild, Quillen, Leibniz cohomologies; Cohomological dimension; Extension; Action; Universal strict general actor; Center; Hochschild; Quillen; Leibniz cohomologies; cohomological dimension; extension; action; universal strict general actor; center},
language = {eng},
number = {1},
pages = {57-78},
title = {Left-right noncommutative Poisson algebras},
url = {http://eudml.org/doc/269479},
volume = {12},
year = {2014},
}

TY - JOUR
AU - José Casas
AU - Tamar Datuashvili
AU - Manuel Ladra
TI - Left-right noncommutative Poisson algebras
JO - Open Mathematics
PY - 2014
VL - 12
IS - 1
SP - 57
EP - 78
AB - 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 special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NPlr-algebras and AWBlr (resp. of NPr-algebras and AWBr) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWBr, the Hochschild or/and Leibniz cohomological dimension of P is ≤ n are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.
LA - eng
KW - Poisson algebra; Algebras with bracket; Leibniz algebra; Representation; Left-right noncommutative Poisson algebra cohomology; Hochschild, Quillen, Leibniz cohomologies; Cohomological dimension; Extension; Action; Universal strict general actor; Center; Hochschild; Quillen; Leibniz cohomologies; cohomological dimension; extension; action; universal strict general actor; center
UR - http://eudml.org/doc/269479
ER -

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