The Dirichlet Hopf Algebra of Arithmetics

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 Hopf algebra 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.

Renormalization as a functor on bialgebras

“…At the mathematical level, the renormalization Hopf algebra found unexpected applications in number theory [21]. Moreover, an intriguing connection was observed between the renormalization bialgebra T (T (B) + ) and a construction involving operads [42,41].…”

“…(21) and (22) are understood in the sense of formal power series in λ. The first term ofā(λ) is Λ(a) = a, which is called the bare Lagrangian in quantum field theory.…”

The Hopf algebra structure of the character rings of classical groups

“…We believe that the general branching scenario is quite universal, and have proposed to take it as a blueprint for quantum field calculations [11,12]. The present work is preparatory to the study of the character ring Hopf algebras at a (conformal) quantum field level using vertex operator techniques.…”

Algebraic random walks in the setting of symmetric functions

“…Under the evolution of the inner and outer product ARWs, these cordinates can be either multiplicatively scaled (inner) or additively augmented (outer), with appropriate probabilities. Remarkably, the primitive λ -ring operations X → X + Y , X → XY at the level of the underlying alphabet of the symmetric function ring, happen to be reflected in the arithmetic operations of addition + and multiplication · on the coordinates of the distribution ρ (see also [20], appendix). (Λ, ∇) .…”