The antipode of of a Com-PreLie Hopf algebra
Lo\"ic Foissy (LMPA)
TL;DR
This paper investigates the relationship between the antipode and preLie product in Com-PreLie Hopf algebras, exemplified by the Connes-Kreimer algebra, and explores how this compatibility aids in computing antipodes using combinatorial coefficients.
Contribution
It introduces a framework for understanding the antipode in Com-PreLie Hopf algebras and applies it to the Connes-Kreimer algebra, revealing new combinatorial insights.
Findings
Established compatibility between antipode and preLie product in Com-PreLie Hopf algebras.
Derived combinatorial coefficients for antipodes using partitions and harmonic sums.
Extended the understanding of the antipode in the Connes-Moscovici subalgebra.
Abstract
We study the compatibility between the antipode and the preLie product of a Com-PreLie Hopf algebra, that is to say a commutative Hopf algebra with a complementary preLie product, compatible with the product and the coproduct in a certain sense. An example of such a Hopf algebra is the Connes-Kreimer Hopf algebra, with the preLie product given by graftingof forests, extending the free preLie product of grafting of rooted trees. This compatibility is then used to study the antipode of the Connes-Moscovici subalgebra, whichcan be defined with the help of this preLie product. The antipode of the generators of this subalgebra gives a family of combinatorial coefficients indexed by partitions,which can be computed with the help of iterated harmonic sums.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Rings, Modules, and Algebras