$q$ -三元组可能代数基的推广
Generalization of the possible algebraic basis of $q$-triplets
摘要：
所谓的$q$ -三联体在2004年被发现，然后在2005年在自然界中被发现。在2005年取得了相关的进一步进展，当时提出了一种可能性，即它们可以反映基于自对偶关系$q \to 2-q$（{ \it加法对偶}）和$q \to 1/q$（{ \it乘法对偶}）的组合的整个无限代数。整个代数坍缩成单个不动点$q=1$，对应于玻尔兹曼-吉布斯熵和统计力学。对于$q \ne 1$，出现了指数$q$的无限集合，原则上对应于可以用所谓的$q$ -统计描述的给定复杂系统的无限数量的物理性质。提出的基本思想是，对于一个给定的普遍性类的系统，一个小的数量（通常是一个或两个）的独立$q$指数存在，无限的其他人从这几个通过简单地使用代数的关系。$q$ -三元组似乎构成了代数的几个中心元素。在过去的十年中，大量的$q$ -三联体在自然、人工和社会系统的分析、计算、实验和观测结果中得到了展示。他们中的一些人确实满足可用的代数构造的加法和乘法对偶，但有些人似乎违反it. In目前的工作中，我们推广这两个对偶，希望更广泛的系统内可以处理。推广的基础由{ \it自对偶}关系$q \to q_a(q) \equiv \frac{(a+2) -aq}{a-(a-2)q} \,\, (a \in {\cal R})$给出。我们验证了$q_a(1)=1$，以及$q_2(q)=2-q$和$q_0(q)=1/q$。物理激励这种推广，我们简要地回顾说明性的应用$q$ -统计，以展示可能的候选人，本广义代数可能是有用的。
英文摘要：
The so called $q$-triplets were conjectured in 2004 and then found in nature in 2005. A relevant further step was achieved in 2005 when the possibility was advanced that they could reflect an entire infinite algebra based on combinations of the self-dual relations $q \to 2-q$ ({\it additive duality}) and $q \to 1/q$ ({\it multiplicative duality}). The entire algebra collapses into the single fixed point $q=1$, corresponding to the Boltzmann-Gibbs entropy and statistical mechanics. For $q \ne 1$, an infinite set of indices $q$ appears, corresponding in principle to an infinite number of physical properties of a given complex system describable in terms of the so called $q$-statistics. The basic idea that is put forward is that, for a given universality class of systems, a small number (typically one or two) of independent $q$ indices exist, the infinite others being obtained from these few ones by simply using the relations of the algebra. The $q$-triplets appear to constitute a few central elements of the algebra. During the last decade, an impressive amount of $q$-triplets have been exhibited in analytical, computational, experimental and observational results in natural, artificial and social systems. Some of them do satisfy the available algebra constructed solely with the additive and multiplicative dualities, but some others seem to violate it. In the present work we generalize those two dualities with the hope that a wider set of systems can be handled within. The basis of the generalization is given by the {\it selfdual} relation $q \to q_a(q) \equiv \frac{(a+2) -aq}{a-(a-2)q} \,\, (a \in {\cal R})$. We verify that $q_a(1)=1$, and that $q_2(q)=2-q$ and $q_0(q)=1/q$. To physically motivate this generalization, we briefly review illustrative applications of $q$-statistics, in order to exhibit possible candidates where the present generalized algebras could be useful.
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