Lie antialgebras: premices

Valentin Ovsienko (1)

1 ICJ - Institut Camille Jordan

The main purpose of this work is to develop the basic notions of the Lie theory for commutative algebras. We introduce a class of $\mathbbZ_2$-graded commutative but not associative algebras that we call ``Lie antialgebras''. These algebras are closely related to Lie (super)algebras and, in some sense, link together commutative and Lie algebras. The main notions we define in this paper are: representations of Lie antialgebras, an analog of the Lie-Poisson bivector (which is not Poisson) and central extensions. We also classify simple finite-dimensional Lie antialgebras.

Mots clés: Graded commutative algebras Lie superalgebras Symplectic geometry

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Algèbres quantiques [math.QA]; Géométrie symplectique [math.SG]

Valentin Ovsienko. Lie antialgebras: premices. 2010. ⟨hal-00145724v6⟩