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 (not necessarily commutative nor associative) ring B and a B-algebra A, a Hasse–Schmidt derivation is a map of B-algebras


 * $$D: A \to A[\![t]\!]$$

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 (defined on, say, Rn) and B=R, the map


 * $$f \mapsto \exp\left(t \frac d {dx}\right) f(x) = f + t \frac {df}{dx} + \frac {t^2}2 \frac {d^2 f}{dx^2} + \cdots$$

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


 * $$\operatorname{NSymm} = \mathbf Z \langle Z_1, Z_2, \ldots \rangle$$

of noncommutative symmetric functions in countably many variables Z1, Z2, ...: the part $$D_i : A \to A$$ of D which picks the coefficient of $$t^i$$, is the action of the indeterminate Zi.

Applications
Hasse–Schmidt derivations on the exterior algebra $A = \bigwedge M$ 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.