Hasse–Schmidt derivation
# Hasse–Schmidt derivation

In mathematics, a Hasse–Schmidt derivation is an extension of the notion of a derivation. The concept was introduced by Schmidt & Hasse (1937).

## 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 Gatto & Salehyan (2016, §3.4), which also contains the following example: for A being the ring of infinitely differentiable functions (defined on, say, $R^n)$ 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

Hazewinkel (2012) 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 $Z_1,$ $Z_2,$ ...: the part $D_{i}:A\to A$ of D which picks the coefficient of $t^{i}$, is the action of the indeterminate $Z_i.$

## Applications

Hasse–Schmidt derivations on the exterior algebra A=⋀M{\textstyle A=\bigwedge M} of some B-module M have been studied by Gatto & Salehyan (2016, §4). Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem. See also Gatto & Scherbak (2015).

