Английская Википедия:Hasse–Schmidt derivation

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In mathematics, a Hasse–Schmidt derivation is an extension of the notion of a derivation. The concept was introduced by Шаблон:Harvtxt.

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>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 Шаблон:Harvtxt, 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>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

Шаблон:Harvtxt shows that a Hasse–Schmidt derivation is equivalent to an action of the bialgebra

<math>\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>D_i : A \to A</math> of D which picks the coefficient of <math>t^i</math>, is the action of the indeterminate Z_i.

Applications

Hasse–Schmidt derivations on the exterior algebra <math display="inline">A = \bigwedge M</math> of some B-module M have been studied by Шаблон:Harvtxt. Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem. See also Шаблон:Harvtxt.

References

• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation
• Шаблон:Citation