%0 Journal Article
%T Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions
%A Michiel Hazewinkel
%J Axioms
%D 2012
%I MDPI AG
%R 10.3390/axioms1020149
%X 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.
%K non-commutative symmetric functions
%K Hasse-Schmidt derivation
%K higher derivation
%K Heerema formula
%K Mirzavaziri formula
%K non-commutative Newton formulas
%U http://www.mdpi.com/2075-1680/1/2/149