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 Der_k(R). This means that all differential operators can be built from functions in R and the derivations in S through repeated application and composition. Examples include the algebra of differential operators on a smooth manifold, where vector fields (derivations of C∞ functions) together with functions generate the full operator algebra; and the Weyl algebra, generated by the coordinate x and the derivative d/dx.

Applications of the derivationsgenerating perspective appear in D-module theory, deformation and microlocal analysis, and algebraic geometry. It provides a practical criterion for constructing differential operator algebras from a manageable set of generators. Limitations arise in singular or non-smooth settings, where derivations alone may not generate higher-order operators without additional structure.

See also: derivation, differential operator, D-module, Weyl algebra, tangent sheaf, Lie algebra.