Английская Википедия:Hasse–Schmidt derivation
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 Zi.
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