Formulas for Birkhoff-(Rota-Baxter) decompositions related to connected bialgebra
Frederic Menous (LM-Orsay)

TL;DR
This paper provides explicit formulas for Birkhoff-(Rota-Baxter) decompositions of characters on connected Hopf algebras, linking them to a universal element in a quasi-shuffle algebra, with applications to quantum field theory renormalization.
Contribution
It introduces formulas for BRB decomposition in the group of characters on connected Hopf algebras using a universal element derived from a quasi-shuffle algebra.
Findings
Established a canonical injective morphism from a connected Hopf algebra to a quasi-shuffle algebra.
Demonstrated that BRB decomposition is governed by a universal element in the quasi-shuffle algebra.
Linked the algebraic structures to the BPHZ renormalization process in quantum field theory.
Abstract
In recent years, The BPHZ algorithm for renormalization in quantum field theory has been interpreted, after dimensional regularization, as the Birkhoff-(Rota-Baxter) decomposition (BRB) of characters on the Hopf algebra of Feynmann graphs, with values in a Rota-Baxter algebra. We give in this paper formulas for the BRB decomposition in the group of characters on a connected Hopf algebra , with values in a Rota-Baxter (commutative) algebra . To do so we first define the stuffle (or quasi-shuffle) Hopf algebra associated to an algebra . We prove then that for any connected Hopf algebra , there exists a canonical injective morphism from to . This morphism induces an action of on so that the BRB decomposition in is determined by the…
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Finite Group Theory Research
