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).

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

[math]\displaystyle{ D: A \to A[\![t]\!] }[/math]

taking values in the ring of formal power series with coefficients in A. This definition is found in several places, such as (Gatto Salehyan), which also contains the following example: for A being the ring of infinitely differentiable functions (defined on, say, Rⁿ) and B=R, the map

[math]\displaystyle{ 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 }[/math]

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

[math]\displaystyle{ \operatorname{NSymm} = \mathbf Z \langle Z_1, Z_2, \ldots \rangle }[/math]

of noncommutative symmetric functions in countably many variables Z₁, Z₂, ...: the part [math]\displaystyle{ D_i : A \to A }[/math] of D which picks the coefficient of [math]\displaystyle{ t^i }[/math], is the action of the indeterminate Z_i.

Applications

Hasse–Schmidt derivations on the exterior algebra [math]\displaystyle{ A = \bigwedge M }[/math] of some B-module M have been studied by (Gatto Salehyan). Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem. See also (Gatto Scherbak).

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 
• Gatto, Letterio; Scherbak, Inna (2015), Remarks on the Cayley-Hamilton Theorem 
• Hazewinkel, Michiel (2012), "Hasse–Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions", Axioms 1 (2): 149–154, 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. 1937 (177): 215–237, doi:10.1515/crll.1937.177.215, ISSN 0075-4102