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Axioms 2012, 1(2), 149-154; doi:10.3390/axioms1020149
Published: 16 July 2012
Abstract: Let NSymm be the Hopf algebra of non-commutative symmetric functions (in an infinity of indeterminates): NSymm=Z . It is shown that an associative algebra A with a Hasse-Schmidt derivation d = (id, d1, d2, ...) on it is exactly the same as an NSymm module algebra. The primitives of NSymm act as ordinary derivations. There are many formulas for the generators Zi in terms of the primitives (and vice-versa). This leads to formulas for the higher derivations in a Hasse-Schmidt derivation in terms of ordinary derivations, such as the known formulas of Heerema and Mirzavaziri (and also formulas for ordinary derivations in terms of the elements of a Hasse-Schmidt derivation). These formulas are over the rationals; no such formulas are possible over the integers. Many more formulas are derivable.
Let A be an associative algebra (or any other kind of algebra for that matter). A derivation on A is an endomorphism of the underlying Abelian group of A such that
A Hasse-Schmidt derivation is a sequence(d0 = id, d1, d2, ..., dn, ...) of endomorphisms of the underlying Abelian group such that for all n ≥ 1.
Note that d1 is a derivation as defined by Equation 1.1. The individual dn that occurs in a Hasse-Schmidt derivation is also sometimes called a higher derivation.
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.
It is the purpose of this short note to show that such formulas follow directly from some easy results about the Hopf algebra NSymm of non-commutative symmetric functions. In fact this Hopf algebra constitutes a universal example concerning the matter.
2. Hopf Algebras and Hopf Module Algebras
Everything will take place over a commutative associative unital base ring k; unadorned tensor products will be tensor products over k. In this note k will be the ring of integers Z, or the field of rational numbers Q.
Recall that a Hopf algebra over k is a k-module H together with five k-module morphisms , , , , such that (H,m,e) is an associative k-algebra with unit, (H,μ,ε) is a co-associative co-algebra with co-unit, μ and ε are algebra morphisms (or, equivalently, that m and e are co-algebra morphisms), and such that ι satisfies , . The antipode ι will play no role in what follows. If there is no antipode (specified) one speaks of a bi-algebra. For a brief introduction to Hopf algebras (and co-algebras) with plenty of examples see Chapters 2 and 3 of .
Recall also that an element is called primitive if . These form a sub-k-module of H and form a Lie algebra under the commutator difference product . I shall use Prim(H) to denote this k-Lie-algebra.
Given a Hopf algebra over k, a Hopf module algebra is a k-algebra A together with an action of the underlying algebra of H on (the underlying module of) A such that:
and where I have used Sweedler-Heynemann notation for the co-product.
Note that this means that the primitive elements of H act as derivations.
3. The Hopf Algebra NSymm of Non-Commutative Symmetric Functions
As an algebra over the integers NSymm is simply the free associative algebra in countably many (non-commuting) indeterminates, . The comultiplication and counit are given by
As NSymm is free as an associative algebra, it is no trouble to verify that this defines a bi-algebra. The seminal paper  started the whole business of non-commutative symmetric functions, and is now a full-fledged research area in its own right.
Now consider an NSymm Hopf module, algebra A. Then, by Equations 2.1 and 3.1 the module endomorphims defined by the actions of the Zn, n ≥ 1 , dn (a) = Zn a, define a Hasse-Schmidt derivation. Conversely, if A is a k-algebra together with a Hasse-Schmidt derivation one defines a NSymm Hopf module algebra structure on A by setting Zn a = dn (a). This works because NSymm is free as an algebra.
Thus an NSymm Hopf module algebra A is precisely the same thing as a k-algebra A together with a Hasse-Schmidt derivation on it and the matter of writing the elements of the sequence of morphisms that make up the Hasse-Schmidt derivation in terms of ordinary derivations comes down to the matter of finding enough primitives of NSymm so that the generators, Zn, can be written as polynomials in these primitives.
4. The Newton Primitives of NSymm
Define the non-commutative polynomials Pn and by the recursion formulas
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 Zn in terms of the P’s or . However, to do that one definitely needs to use rational numbers and not just integers . For instance
The key observation is now:
The elements Pn and are primitive elements of the Hopf algebra NSymm.
The proof is a straightforward uncomplicated induction argument using the recursion Formulas 4.1. See e.g., , page 147.
Using the an immediate corollary is the following main theorem from .
Let A be an associative algebra over the rational numbers Q and let be a Hasse-Schmidt derivation on it. Then the δn defined recursively by
are ordinary derivations and
the formulas expressing the Zn in terms of the are unique and so denominators are really needed.
4.9. Comment and Example
There are many more primitive elements in NSymm than just the and Pn. One could hope that by using all of them, integral formulas for the Zn in terms of primitives would become possible. This is not the case. The full Lie algebra of primitives of NSymm was calculated in . It readily follows from the description there that , the sub-algebra of NSymm generated by all primitive elements is strictly smaller than NSymm. In fact much smaller in a sense that is specified in locus citandi. Thus the theorem does not hold over the integers.
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 NSymm itself, as follows: The Hopf algebra NSymm is graded by giving Zn degree n. Note that each graded piece is a free Z-module of finite rank. Let QSymm, often called the Hopf algebra of quasi-symmetric functions, be the graded dual Hopf algebra. Then each Zn defines a functional . Now define an endomorphism dn of QSymm as the composed morphism
Then the dn form a Hasse-Schmidt derivation of which the components cannot be written as integer polynomials in ordinary derivations.
5. The Hopf Algebra LieHopf
In  a formula for manufacturing Hasse-Schmidt derivations from a collection of ordinary derivations is shown that is more pleasing—at least to me—than 4.6. This result from locus citandi can be strengthened to give a theorem similar to Theorem 4.4 but with more symmetric formulae. This involves another Hopf algebra over the integers which I like to call LieHopf.
As an algebra LieHopf is again the free associative algebra in countably many indeterminates . However, this time the co-multiplication and co-unit are defined by
so that all the Un are primitive. Also, in fact the Lie algebra of primitives of this Hopf algebra is the free Lie algebra on countably many generators.
Over the integers LieHopf and NSymm are very different but over the rationals they become isomorphic. There are very many isomorphisms. A particularly nice one is given in considering the power series identity
which gives the following formulae for the U’s in terms of the Z’s and vice versa.
For two detailed proofs that these formulas do indeed give an isomorphism of Hopf algebras see ; or see Chapter 6 of . In terms of derivations, reasoning as above in Section 4, this gives the following theorem.
Let A be an algebra over the rationals and let be a Hasse-Schmidt derivation on it. Then the defined by
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  with which I started this section.
Hasse-Schmidt derivations on an associative algebra A are exactly the same as Hopf module algebra structures on A for the Hopf algebra NSymm. This leads to formulas connecting ordinary derivations to higher derivations.
It remains to explore this phenomenon for other kinds of algebras.
The dual of NSymm is QSymm, the Hopf algebra of quasi-symmetric functions. It remains to be clarified what a coalgebra comodule over QSymm means in terms of coderivations. There are also other (mixed) variants to be further explored.
References and Notes
- Heerema, N. Higher derivations and automorphisms. Bull. American Math. Soc. 1970, 1212–1225, doi:10.1090/S0002-9904-1970-12609-X.
- Mirzavaziri, M. Characterization of higher derivations on algebras. Comm. Algebra 2010, 38, 981–987, doi:10.1080/00927870902828751.
- Hazewinkel, M.; Nadiya, G.; Vladimir, V.K. Algebras, Rings, and Modules:Lie Algebras and Hopf Algebras.; American Mathematicial Society: Providence, RI, USA, 2010.
- Gel’fand, I.M.; Daniel, K.; Alain, L.; Bernard, L.; Vladimir, S.R.; Jean-Yves, T. Noncommutative symmetric functions. Adv. Math. 1995, 112, 218–348.
- 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.
- Hazewinkel, M. The primitives of the Hopf algebra of noncommutative symmetric functions. São Paulo J. Math. Sci. 2007, 1, 175–203.
- Hazewinkel, M. The Leibniz Hopf Algebra and Lyndon Words; CWI: Amsterdam, The Netherlands, 1996.
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