Hasse-Schmidt derivations and the Hopf algebra of noncommutative symmetric functions

Abstract

Let A be an associative algebra (or any other kind of algebra for that matter). A derivation on A is an endomorphism \del of the underlying Abelian group of A such that \del(ab)=a(\del b)+(\del a)b for all a,b\in A (1.1) A Hasse-Schmidt derivation is a sequence (d_0=id,d_1,d_2,...,d_n,...) of endomorphisms of the underlying Abelian group such that for all n \ge 1 d_n(ab)= \sum_{i=0}^n (d_ia)(d_{n-i}b) (1.2) Note that d_1 is a derivation as defined by (1.1). The individual d_n that occur in a Hasse-Schmidt derivation are also sometimes called higher derivations. A question of some importance is whether Hasse-Schmidt derivations can be written down in terms of polynomials in ordinary derivations. For instance in connection with automatic continuity for Hasse-Schmidt derivations on Banach algebras. Such formulas have been written down by, for instance, Heerema and Mirzavaziri in [5] and [6]. They also will be explicitly given below. It is the purpose of this short note to show that such formulas follow directly from some easy results about the Hopf algebra NSymm of non-commutative symmetric functions. In fact this Hopf algebra constitutes a universal example concerning the matter.