Next Article in Journal / Special Issue
Quasitriangular Structure of Myhill–Nerode Bialgebras
Previous Article in Journal / Special Issue
Valued Graphs and the Representation Theory of Lie Algebras
Axioms 2012, 1(2), 149-154; doi:10.3390/axioms1020149
Communication

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

Burg. ‘s Jacob Laan 18, NL-1401BR BUSSUM, The Netherlands
Received: 4 May 2012 / Revised: 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)
View Full-Text   |   Download PDF [194 KB, 20 July 2012; original version 16 July 2012]

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 non-commutative symmetric functions; Hasse-Schmidt derivation; higher derivation; Heerema formula; Mirzavaziri formula; non-commutative Newton formulas
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
SciFeed

Share & Cite This Article

Further Mendeley | CiteULike
Export to BibTeX |
EndNote |
RIS
MDPI and ACS Style

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

View more citation formats

Related Articles

Article Metrics

Comments

[Return to top]
Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert