  In this paper we construct a graded Lie algebra on the space of cochains on a
$\mathbbZ_2$-graded vector space that are skew-symmetric in the odd variables.
The Lie bracket is obtained from the classical Gerstenhaber bracket by
(partial) skew-symmetrization; the coboundary operator is a skew-symmetrized
version of the Hochschild differential. We show that an order-one element $m$
satisfying the zero-square condition $[m,m]=0$ defines an algebraic structure
called "Lie antialgebra". The cohomology (and deformation) theory of these
algebras is then defined. We present two examples of non-trivial cohomology
classes which are similar to the celebrated Gelfand-Fuchs and Godbillon-Vey
classes.
