Abstract We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.

中文翻译：

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分解代数和轴向代数

摘要 我们引入分解代数作为轴代数、Majorana 代数和 Griess 代数的自然推广。它们弥补了轴向代数的三个局限性：（1）它们将融合律与域中的特定值分开，从而允许重复特征值；(2) 它们允许不是由幂等乘法产生的分解；(3) 他们承认同态的自然概念，使它们成为一个很好的范畴。我们利用这些事实来加强轴代数和群之间的联系。特别是，我们提供了一个通用宫本群的定义，它在一些温和的假设下使这种连接具有函子性。我们通过解释表示理论和关联方案如何帮助为给定（置换）群构建分解代数来说明我们的理论。这种结构导致了大量的例子。我们还借此机会修复了这个快速扩展的主题中的一些术语。