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Let _{1}, _{2}, ...) on it is exactly the same as an _{i}

Let

A Hasse-Schmidt derivation is a sequence(_{0} = _{1}, _{2}, ..., _{n}, ...) of endomorphisms of the underlying Abelian group such that for all

Note that _{1} is a derivation as defined by Equation 1.1. The individual _{n}

A question of some importance is whether Hasse-Schmidt derivations can be written down in terms of polynomials in ordinary derivations. For instance, in connection with automatic continuity for Hasse-Schmidt derivations on Banach algebras.

Such formulas have been written down by, for instance, Heerema and Mirzavaziri in [

It is the purpose of this short note to show that such formulas follow directly from some easy results about the Hopf algebra

Everything will take place over a commutative associative unital base ring

Recall that a Hopf algebra over

Recall also that an element

Given a Hopf algebra over

and where I have used Sweedler-Heynemann notation for the co-product.

Note that this means that the primitive elements of

As an algebra over the integers

As

Now consider an _{n}_{n }_{n} a_{n }a_{n }

Thus an _{n}

Define the non-commutative polynomials _{n}

These are non-commutative analogues of the well known Newton formulas for the power sums in terms of the complete symmetric functions in the usual commutative theory of symmetric functions. It is not difficult to write down an explicit expression for these polynomials:

Nor is it difficult to write down a formula for the _{n}

The key observation is now:

The elements _{n}

The proof is a straightforward uncomplicated induction argument using the recursion Formulas 4.1. See e.g., [

Using the

Let _{n}

are ordinary derivations and

Where

Because

the formulas expressing the _{n}

There are many more primitive elements in _{n}_{n}

A concrete example of a Hasse-Schmidt derivation of which the constituting endomorphisms cannot be written as integral polynomials in derivations can be given in terms of _{n}_{n}_{n}

Then the _{n}

In [

As an algebra

so that all the _{n}

Over the integers

which gives the following formulae for the

For two detailed proofs that these formulas do indeed give an isomorphism of Hopf algebras see [

Let

are (ordinary) derivations and

Perhaps I should add that for any given collection of ordinary derivations, Formula 5.7 yields a Hasse-Schmidt derivation. That is the theorem from [

Hasse-Schmidt derivations on an associative algebra

It remains to explore this phenomenon for other kinds of algebras.

The dual of

This is an instance where the noncommutative formulas are more elegant and also easier to prove than their commutative analogues. In the commutative case there are all kinds of multiplicities that mess things up.