Deformations and geometric rigidity of Leibniz algebras

A formal deformation of a Leibniz algebra A over a commutative ring k, consists of a formal series F = Sigma(n=0)(+infinity)f(n)t(n), extending the Leibniz bracket of A, defined by f(0). We show that if f is a 2-cocycle for the cohomology of A with coefficients in its adjoint representation, a sufficient condition for the existence of F with f(1) = f, is the vanishing of the third cohomology group of A with coefficients in its adjoint representation. In the last section, we study deformations from the point of view of algebraic varieties.