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The Dirichlet Hopf algebra of arithmetics

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Fauser, B and Jarvis, PD (2007) The Dirichlet Hopf algebra of arithmetics. Journal of Knot Theory and Its Ramifications, 16 (4). pp. 379-438. ISSN 0218-2165

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Abstract

Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the "Dirichlet Hopf algebra of arithmetics" by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopfalgebra is then cured by introducing an "unrenormalized" coproduct and an "unrenormalized" pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra (literally: antipodal convolution). This can be modelled alternatively by employing Rota–Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

Item Type: Article
Journal or Publication Title: Journal of Knot Theory and Its Ramifications
Publisher: World Scientific Publishing Company Pte Ltd
Page Range: pp. 379-438
ISSN: 0218-2165
Identification Number - DOI: 10.1142/S0218216507005269
Date Deposited: 07 Apr 2008 14:41
Last Modified: 18 Nov 2014 03:36
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