title: Formal groups andZ-entropies.
abstract: We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them theZ-entropies. Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that everyZ-entropy iscomposable(Tempesta 2016Ann. Phys.365, 180-197. (doi:10.1016/j.aop.2015.08.013)). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon-Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, theZ-entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.
authors.affiliation: Departamento de Física Teórica II (Métodos Matemáticos de la Física), Facultad de Físicas, Universidad Complutense de Madrid, 28040 Madrid, Spain; Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, No. 13-15, 28049 Madrid, Spain.
authors.forename: Piergiulio
authors.identifier: 0000-0002-5186-2378
authors.initials: P
authors.lastname: Tempesta
keywords: generalized entropies; group theory; information theory