Hasse–Schmidt derivation

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

$D:A\to A[[t]]$

taking values in the ring of formal power series with coefficients in A. This definition is found in several places, such as Template: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

$f\mapsto \exp \left(t\frac{d}{dx}\right)f(x)=f+t\frac{df}{dx}+\frac{t^2}{2}\frac{d^{2}f}{d{x}^2}+\cdots$

is a Hasse–Schmidt derivation, as follows from applying the Leibniz rule iteratedly.

Equivalent characterizations

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

$\mathrm{NSymm}=𝐙⟨Z_1,Z_2,\dots ⟩$

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 $A=\bigwedge M$ of some B-module M have been studied by Template:Harvtxt. Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem. See also Template:Harvtxt.

References

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