derivationsgenerating
Derivationsgenerating is a term used to describe the situation in algebra and geometry where a collection of derivations on a ring, algebra, or geometric space suffices to generate the full algebra of differential operators on that object. A derivation is a k-linear map D from a k-algebra R to itself that satisfies the Leibniz rule D(ab) = a D(b) + b D(a). Given a subset S of Der_k(R), the derivationsgenerating property means that the smallest subalgebra of the differential operator algebra D_R containing R and S is the whole D_R (or, in practice, that S generates the relevant portion of D_R under addition and composition).
In smooth or regular contexts, a standard result is that D_R is generated as an R-algebra by
Applications of the derivationsgenerating perspective appear in D-module theory, deformation and microlocal analysis, and algebraic geometry.
See also: derivation, differential operator, D-module, Weyl algebra, tangent sheaf, Lie algebra.