Hopf Algebras of Formal Diffeomorphisms and Numerical Integration on Manifolds
نویسندگان
چکیده
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 aimed at studying Lie-group integrators for differential equations evolving on manifolds. Lie-group integrators are based on general Lie group actions on a manifold, and classical Runge–Kutta integrators appear in this setting as the special case of R acting upon itself by translations. Lie–Butcher theory combines classical B-series on R with Lie-series on manifolds. The underlying Hopf algebra HN combines the Connes–Kreimer Hopf algebra with the shuffle Hopf algebra of free Lie algebras. Aimed at a general mathematical audience, we give an introduction to Hopf algebraic structures and their relationship to structures appearing in numerical analysis. In particular we explore the close connection between Lie series, time-dependent Lie series and Lie–Butcher series for diffeomorphisms on manifolds. The role of the Euler and Dynkin idempotents in numerical analysis is discussed. A non-commutative version of a Faà di Bruno bialgebra is introduced, and the relation to non-commutative Bell polynomials is explored. 1 ar X iv :0 90 5. 00 87 v1 [ m at h. N A ] 1 M ay 2 00 9
منابع مشابه
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 ...
متن کاملGroups of tree-expanded series
In [5, 6] we introduced three Hopf algebras on planar binary trees related to the renormalization of quantum electrodynamics. One of them, the algebra H, is commutative, and is therefore the ring of coordinate functions of a proalgebraic group G. The other two algebras, H and H , are free non-commutative. Therefore their abelian quotients are the coordinate rings of two proalgebraic groups G an...
متن کاملK-invariants of conjugacy classes of pseudo-Anosov diffeomorphisms and hyperbolic 3-manifolds
New invariants of 3-dimensional manifolds appearing in the Ktheory of certain operator algebras are introduced. First, we consider the conjugacy problem for pseudo-Anosov diffeomorphisms of a compact surface X. The operator algebra in question is an AF -algebra attached to stable (unstable) foliation of the pseudo-Anosov diffeomorphism. We prove that conjugacy classes of commensurable pseudoAno...
متن کاملLessons from Quantum Field Theory Hopf Algebras and Spacetime Geometries
We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge...
متن کامل