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

• Main Author: Michiel Hazewinkel
• Format: Article
• Language: English
• Published: MDPI AG 2012-07-01
• Series: Axioms
• Subjects: non-commutative symmetric functions; Hasse-Schmidt derivation; higher derivation; Heerema formula; Mirzavaziri formula; non-commutative Newton formulas
• Online Access: http://www.mdpi.com/2075-1680/1/2/149
• ISSN: 2075-1680

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.