Article

# The primitives of the Hopf algebra of noncommutative symmetric functions

11/2004;

Source: arXiv

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**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.10/2011; - [Show abstract] [Hide abstract]

**ABSTRACT:**It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions, Symm. We offer the cohomology of the loop space of the suspension of the infinite complex projective space as a topological model for the ring of quasisymmetric functions, QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on the loop space that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU. The canonical Thom spectrum over the loops on the suspension of BU(1) is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.05/2006; -
##### Article: Witt vectors. Part 1

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**ABSTRACT:**This is the first part of a 2 part survey on the functor of the big and p-adic Witt vectors.05/2008;

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