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On the Hopf Algebraic Structure of Lie Group Integrators | | H. Z. Munthe-Kaas
; W. M. Wright
; PostScript
; PDF
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; | | Date: |
1 Mar 2006 | | Subject: | Commutative Algebra; Numerical Analysis | | Abstract: | A commutative but not cocommutative graded Hopf algebra $Hn$, based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees $Hc$, developed by Butcher in his study of Runge--Kutta methods and later rediscovered by Connes and Moscovici in the context of non-commutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that $Hn$ is naturally obtained from a universal object in a category of non-commutative derivations, and in particular, it forms a foundation for the study of numerical integrators based on non-commutative Lie group actions on a manifold. Recursive and non-recursive definitions of the coproduct and the antipode are derived. It is also shown that the dual of $Hn$ is a Hopf algebra of Grossman and Larson. $Hn$ contains two well-known Hopf algebras as special cases: The Hopf algebra $Hc$ of Butcher--Connes--Kreimer is identified as a proper subalgebra of $Hn$ using the image of a tree symmetrization operator. The Hopf algebra $Hf$ of the Free Associative Algebra is obtained from $Hn$ by a quotient construction. | | Source: | arXiv, math/0603023 | | Services: | Forum | Review | PDF | Favorites | |
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