From Poisson algebras to Gerstenhaber algebras

Kosmann-Schwarzbach, Yvette

Annales de l'Institut Fourier, Tome 46 (1996) no. 5, pp. 1243-1274.

On montre que l’on peut construire un crochet de Poisson pair à partir d’un crochet de Gerstenhaber, à l’aide d’une dérivation impaire de carré nul, dans la catégorie des algèbres de Loday (algèbres munies d’un crochet non antisymétrique, généralisant les crochets de Lie, appelées jusqu’à présent dans la littérature, algèbres de Leibniz). Ces “crochets dérivés” donnent des crochets de Lie sur certains quotients, et sur certaines sous-algèbres abéliennes. On peut expliquer ainsi l’origine du crochet de Lie sur l’espace des formes différentielles co-exactes sur une variété de Poisson. Nous étudions les crochets dérivés sur l’espace des cochaînes sur une algèbre associative ou de Lie. Enfin nous relions les résultats précédents à diverses généralisations de la notion d’algèbre de Batalin-Vilkovisky.

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.

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