I will construct a graded Lie algebra on the space of cochains on a $Z_2$-graded vector space, skew-symmetric in the odd variables. The Lie bracket is obtained from the classical Gerstenhaber bracket by (partial) skew-symmetrization; the coboundary operator is an amazing mixture of the Hochschild and Chevalley-Eilenberg differentials. I will show that an order-one element $m$ satisfying the zero-square condition defines an algebraic structure called ``Lie antialgebra´´, I will explain geometric origins of these algebras. Two examples of non-trivial cohomology classes will be constructed, these classes are similar to the celebrated Gelfand-Fuchs and Godbillon-Vey classes.