Axioms 2012, 1(2), 149-154; doi:10.3390/axioms1020149
Communication

Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions

Received: 4 May 2012; in revised form: 25 June 2012 / Accepted: 25 June 2012 / Published: 16 July 2012
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract: Let NSymm be the Hopf algebra of non-commutative symmetric functions (in an infinity of indeterminates): . It is shown that an associative algebra A with a Hasse-Schmidt derivation ) 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 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.
Keywords: non-commutative symmetric functions; Hasse-Schmidt derivation; higher derivation; Heerema formula; Mirzavaziri formula; non-commutative Newton formulas
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MDPI and ACS Style

Hazewinkel, M. Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions. Axioms 2012, 1, 149-154.

AMA Style

Hazewinkel M. Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions. Axioms. 2012; 1(2):149-154.

Chicago/Turabian Style

Hazewinkel, Michiel. 2012. "Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions." Axioms 1, no. 2: 149-154.

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