Lie antialgebras: Prémices

The main purpose of this work is to develop the basic notions of the Lie theory for commutative algebras. We introduce a class of Z2-graded commutative but not associative algebras that we call “Lie antialgebras”. These algebras form a very special class of Jordan superalgebras, they 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 will explain the geometric origins of Lie antialgebras and provide a number of examples. We also classify simple finite-dimensional Lie antialgebras. This paper is a new version of the unpublished preprint Ovsienko [10].