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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; 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)
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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 StyleHazewinkel M. Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions. Axioms. 2012; 1(2):149-154.
Chicago/Turabian StyleHazewinkel, Michiel. 2012. "Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions." Axioms 1, no. 2: 149-154.