Hasse–Schmidt derivation

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 . ==References==