Lie antialgebras

Valentin Ovsienko (1)

1 ICJ - Institut Camille Jordan

We introduce a new class of algebras that we call Lie antialgebras. The structure of a Lie antialgebra is defined by a changing parity operation on a $\mathbb{Z}_2$-graded space satisfying a set of identities amazingly ``opposite'' to those of a Lie superalgebra. We classify Lie antialgebras of rank 1 and consider several examples of simple Lie antialgebras. We also define an analog of the Lie-Poisson structure on the space dual to a Lie antialgebra and study the notion of central extensions.

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

Valentin Ovsienko. Lie antialgebras. 2007. ⟨hal-00145724v1⟩