Moyal and Rankin-Cohen deformations of algebras

It is proven that Rankin-Cohen brackets form an associativedeformation of the algebra of polynomials whose coeffcients are holomorphicfunctions on the upper half-plane.

VALUED DEFORMATIONS OF ALGEBRAS

We develop a notion of deformations using a valuation ring as a ring of coefficients. This allows us in particular to consider the classical Gerstenhaber deformations of associative or Lie algebras as valued deformations and to solve the equation of deformations in a polynomial framework. We consider also deformations of the enveloping algebra of a rigid Lie algebra and we define valued deformations for some classes of non associative algebras.