From Poisson algebras to Gerstenhaber algebras

Yvette Kosmann-Schwarzbach

Annales de l'institut Fourier (1996)

  • Volume: 46, Issue: 5, page 1243-1274
  • ISSN: 0373-0956

Abstract

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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 the space of co-exact differential forms on a Poisson manifold. We further examine the derived brackets on the space of cochains of an associative or Lie algebra. Finally, we relate the previous result to various generalizations of the notion of BV-algebra.

How to cite

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Kosmann-Schwarzbach, Yvette. "From Poisson algebras to Gerstenhaber algebras." Annales de l'institut Fourier 46.5 (1996): 1243-1274. <http://eudml.org/doc/75211>.

@article{Kosmann1996,
abstract = {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 the space of co-exact differential forms on a Poisson manifold. We further examine the derived brackets on the space of cochains of an associative or Lie algebra. Finally, we relate the previous result to various generalizations of the notion of BV-algebra.},
author = {Kosmann-Schwarzbach, Yvette},
journal = {Annales de l'institut Fourier},
keywords = {graded Lie algebras; Poisson algebras; Gerstenhaber algebras; Loday algebras; Poisson calculus; Koszul bracket; Vinogradov bracket; cohomology of associative algebras; generalized BV-algebras; Leibniz algebras; cohomology of Lie algebras},
language = {eng},
number = {5},
pages = {1243-1274},
publisher = {Association des Annales de l'Institut Fourier},
title = {From Poisson algebras to Gerstenhaber algebras},
url = {http://eudml.org/doc/75211},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Kosmann-Schwarzbach, Yvette
TI - From Poisson algebras to Gerstenhaber algebras
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 5
SP - 1243
EP - 1274
AB - 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 the space of co-exact differential forms on a Poisson manifold. We further examine the derived brackets on the space of cochains of an associative or Lie algebra. Finally, we relate the previous result to various generalizations of the notion of BV-algebra.
LA - eng
KW - graded Lie algebras; Poisson algebras; Gerstenhaber algebras; Loday algebras; Poisson calculus; Koszul bracket; Vinogradov bracket; cohomology of associative algebras; generalized BV-algebras; Leibniz algebras; cohomology of Lie algebras
UR - http://eudml.org/doc/75211
ER -

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