Hasse–Schmidt derivation

In mathematics, a Hasse–Schmidt derivation is an extension of the notion of a derivation. The concept was introduced by.

Definition

For a ring B and a B-algebra A, a Hasse–Schmidt derivation is a map of B-algebras
taking values in the ring of formal power series with coefficients in A. This definition is found in several places, such as, which also contains the following example: for A being the ring of infinitely differentiable functions and B=R, the map
is a Hasse–Schmidt derivation, as follows from applying the Leibniz rule iteratedly.

Equivalent characterizations

shows that a Hasse–Schmidt derivation is equivalent to an action of the bialgebra
of noncommutative symmetric functions in countably many variables Z₁, Z₂,...: the part of D which picks the coefficient of, is the action of the indeterminate Z_i.

Applications

Hasse–Schmidt derivations on the exterior algebra of some B-module M have been studied by. Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem. See also.