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

Definition

For a (not necessarily commutative nor associative)

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''B'' and a ''B''-

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

''A'', a Hasse–Schmidt derivation is a map of ''B''-algebras :

D: A \to A![t!">.html" ;"title="![t">![t!/math>

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taking values in the ring of formal power series with coefficients in ''A''. This definition is found in several places, such as , 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 \right) f(x) = f + t \frac  + \frac 2 \frac  + \cdots

is a Hasse–Schmidt derivation, as follows from applying the

Leibniz rule Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: * Product rule in differential calculus * General Leibniz rule, a generalization of the product rule * Leibniz integral rule * The alternating series test, al ...

iteratedly.

Equivalent characterizations

shows that a Hasse–Schmidt derivation is equivalent to an action of the

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\operatorname = \mathbf Z \langle Z_1, Z_2, \ldots \rangle

noncommutative symmetric function In mathematics, the noncommutative symmetric functions form a Hopf algebra NSymm analogous to the Hopf algebra of symmetric functions. The Hopf algebra NSymm was introduced by Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vladimir ...

s 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 = \bigwedge M

of some ''B''-module ''M'' have been studied by . Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem. See also .

References

* * * * {{DEFAULTSORT:Hasse-Schmidt derivation Abstract algebra