Non-commutative Hopf algebra of formal diffeomorphisms

Non-commutative Hopf algebra of formal diffeomorphisms

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with the group law being composition of series. The motivation to introduce these Hopf algebras comes ...

A combinatorial non-commutative Hopf algebra of graphs

A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the...

Hopf Algebras of Formal Diffeomorphisms and Numerical Integration on Manifolds

B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge–Kutta methods. Connections to renormalization have been established in recent years. The algebraic structure of classical Runge–Kutta methods is described by the Connes–Kreimer Hopf algebra. Lie–Butcher theory is a generalization of B-series ai...

Non-commutative combinatorial algebra

Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions

Let NSymm be the Hopf algebra of non-commutative symmetric functions (in an infinity of indeterminates): NSymm=Z 1 2 , ,... Z Z   . It is shown that an associative algebra A with a Hasse-Schmidt derivation 1 2 ( , , ,...) d id d d  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 Zi in terms ...