  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.
