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 homomorphism 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 (defined on, say, Rn) 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 Zi.
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 .
References