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.
References
[1]
Heerema, N. Higher derivations and automorphisms. Bull. American Math. Soc. 1970, 1212–1225, doi:10.1090/S0002-9904-1970-12609-X.
[2]
Mirzavaziri, M. Characterization of higher derivations on algebras. Comm. Algebra 2010, 38, 981–987, doi:10.1080/00927870902828751.
[3]
Hazewinkel, M.; Nadiya, G.; Vladimir, V.K. Algebras, Rings, and Modules:Lie Algebras and Hopf Algebras.; American Mathematicial Society: Providence, RI, USA, 2010.
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.
[6]
Hazewinkel, M. The primitives of the Hopf algebra of noncommutative symmetric functions. S？o Paulo J. Math. Sci. 2007, 1, 175–203.
[7]
Hazewinkel, M. The Leibniz Hopf Algebra and Lyndon Words; CWI: Amsterdam, The Netherlands, 1996.
