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, 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

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 Z₁, Z₂, ...: 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 ${\textstyle A=\bigwedge M}$ 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).

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References

Gatto, Letterio; Salehyan, Parham (2016), Hasse–Schmidt derivations on Grassmann algebras, Springer, doi:10.1007/978-3-319-31842-4, ISBN 978-3-319-31842-4, MR 3524604
Gatto, Letterio; Scherbak, Inna (2015), Remarks on the Cayley-Hamilton Theorem, arXiv:1510.03022
Hazewinkel, Michiel (2012), "Hasse–Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions", Axioms, 1 (2): 149–154, arXiv:1110.6108, doi:10.3390/axioms1020149
Schmidt, F.K.; Hasse, H. (1937), "Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von F.K. Schmidt in Jena)", J. Reine Angew. Math., 177: 215–237, doi:10.1515/crll.1937.177.215, ISSN 0075-4102, MR 1581557

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