On the Hopf Algebraic Structure of Lie Group Integrators

Abstract

A commutative but not cocommutative graded Hopf algebra H N , based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees H C , developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that H N is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between H N and four other Hopf algebras is discussed. The dual of H N is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra H C of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of H N using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra H Sh is obtained from H N by a quotient construction. The Hopf algebra H P of ordered trees by Foissy differs from H N in the definition of the product (noncommutative Date concatenation for H P and shuffle for H N ) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.

Stochastic expansions and Hopf algebras

“…The use of Hopf shuffle algebras in stochastic expansions can be traced through, for example, Strichartz (1987), Reutenauer (1993), Gaines (1994), Li & Liu (2000), Kawski (2002), Baudoin (2004), Ebrahimi-Fard & Guo (2006), Murua (2006), Lyons et al (2007) and Manchon & Paycha (2007), to name a few. The paper by Munthe-Kaas & Wright (2008) on the Hopf algebraic of Lie group integrators actually instigated the Hopf algebra direction adopted here. A useful outline on the use of Hopf algebras in numerical analysis can be found therein, as well as the connection to the work by Connes & Miscovici (1998) and Connes & Kreimer (1998) in renormalization in perturbative quantum field theory.…”

Efficient Strong Integrators for Linear Stochastic Systems

“…Hence in both cases the quadrature effort is proportional to QN . Implicit in the relations (8.1) is the natural underlying shuffle algebra created by integration by parts (see Gaines [20,19], Kawksi [32] and Munthe-Kaas and Wright [48]). Two further results are of interest.…”

Magnus expansions and beyond

“…The fundamental importance of algebras of rooted trees and their connections with the representation of the renormalisation group of quantum field theory has been recognised in (Connes & Kreimer 1998, Connes & Kreimer 2000, and this has led to a great deal of interest and further research, not least on geometric integrators: cf. for example (Munthe-Kaas & Wright 2008, Murua 2006.…”