Moufang Patterns and Geometry of Information
Moufang Patterns and Geometry of Information
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DOI：
10.4310/pamq.2023.v19.n1.a7
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发表时间：
2021-07
期刊：
影响因子：
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通讯作者：
N. Combe;Y. Manin;M. Marcolli
中科院分区：
文献类型：
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作者：
N. Combe;Y. Manin;M. Marcolli
数据采集和信息传输技术是基于各种数学模型的编码。“信息几何“一词指的是这样的模型，而“Moufang模式“一词指的是在这样的模型中自然出现的各种复杂的对称性。本文证明了概率分布空间的对称性，赋予它们的信息几何的标准黎曼度量，具有交换Moufang环的结构。我们还证明了概率分布空间上的F-流形结构可以用微分3-网和Malcev代数来描述。然后，我们提出了一个新的建设（非交换）Moufang循环与几乎辛结构在有限域上，然后使用构建一类新的代码循环与相关的量子纠错码和网络的完美张量。
Technology of data collection and information transmission is based on various mathematical models of encoding. The words"Geometry of information"refer to such models, whereas the words"Moufang patterns"refer to various sophisticated symmetries appearing naturally in such models. In this paper we show that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop. We also show that the F-manifold structure on the space of probability distribution can be described in terms of differential 3-webs and Malcev algebras. We then present a new construction of (noncommutative) Moufang loops associated to almost-symplectic structures over finite fields, and use then to construct a new class of code loops with associated quantum error-correcting codes and networks of perfect tensors.