GENERALIZED DEFORMATIONS, KOSZUL RESOLUTIONS, MOYAL PRODUCTS

We generalise Gerstenhaber's theory of deformations, by dropping the a ssumption that the deformation parameter should commute with the eleme nts of the original algebra. We give the associated cohomology and con struct a Koszul resolution for the polynomial algebra C[X] in the ''ho mogeneous'' case. We then develop examples in the case of C[x,y] and f ind some Moyal-like products of a new type. Finally, we show that, for any field K, matrix algebras with coefficients in K and finite degree extensions of K are rigid, as in the commutative case.