算术的Dirichlet Hopf代数
The Dirichlet Hopf algebra of arithmetics
摘要：
在这项工作中，我们系统地发展了“Dirichlet Hopf代数的算术”对偶加法和乘法映射。我们研究的添加剂和乘法反足卷积，未能产生的Hopf代数结构，只服从一个削弱（乘法）同态公理。削弱的结构，称为霍普夫gebra，例如上同调的后果进行了探讨。这一特点的乘法性与完全乘法性的数论算术函数。不被一个霍普夫代数的不足之处，然后通过引入一个“unrenormalized”的余积和一个“unrenormalized”配对治愈。然后，有人认为，正是非互质整数的同态属性（完全乘法）的失败是一个蓝图的问题，在量子场论（QFT）导致需要重整化。重整化是从代数上可靠的Hopf代数到物理上和数论上有意义的Hopf代数的态射。这可以通过使用Rota-Baxter算子来替代地建模。我们强调需要一个无特征的发展，在可能的情况下，有一个良好的起点，概括的代数结构。最后一节提供了三个关键的应用：对称函数理论，量子（矩阵）力学，以及QFT中重整化的组合学，这些可以被认为是从数论水平上的发展中继承下来的。因此，数论函数在QFT中的出现变得很自然。
英文摘要：
In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, obeying only a weakened (multiplicative) homomorphism axiom. The consequences of the weakened structure, called a Hopf gebra, e.g. on cohomology are explored. 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. 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.
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