Valued deformations of algebras

We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve the equation of deformations in a polynomial frame. We consider also the deformations of the enveloping algebra of a rigid Lie algebra and we define valued deformations for some classes of non associative algebras. Table of contents : 1. Valued deformations of Lie algebras 2. Decomposition of valued deformations 3. Deformations of the enveloping algebra of a rigid Lie algebra 4. Deformations of non associative algebras 1 Valued deformations of Lie algebras 1.1 Rings of valuation We recall briefly the classical notion of ring of valuation. Let F be a (commutative) field and A a subring of F. We say that A is a ring of valuation of F if A is a local integral domain satisfying: If x ∈ F −A, then x ∈ m. where m is the maximal ideal of A. A ring A is called ring of valuation if it is a ring of valuation of its field of fractions. corresponding author: e-mail: [email protected] [email protected]. Partially supported by a grant from the Institut of mathematics Simon Stoilow of the Romanian academy. Bucharest