This paper presents the classification over the fields of real and complex numbers, of the minimal ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie algebras and Lie superalgebras spanned by four generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Poincaré superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded (super)algebras. We mention, in particular, the notion of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further by-product, we point out that, contrary to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded superalgebras, a theory invariant under a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

中文翻译：

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最小Z2×Z2分级李（超）代数的分类及一些应用

本文介绍了实数和复数域的分类，最小的 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$- 分级李代数和李超代数由四个发生器跨越并且没有空分级扇区。不等价分级李（超）代数是通过求解由各自分级雅可比恒等式施加的约束获得的。这个数学结果的动机是系统地研究动态系统在分级（超）代数下不变的性质。最近的作品只关注一维的特例${\mathbb{Z}}_2\times {\mathbb{Z}}_2$- 分级庞加莱超代数。作为应用程序，我们能够将最初为这个特殊超代数引入的某些构造扩展到其他列出的${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-分级（超）代数。我们特别提到了${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-分级超空间和不变动力系统（经典世界线西格玛模型和量子哈密顿模型）。作为进一步的副产品，我们指出，与${\mathbb{Z}}_2\times {\mathbb{Z}}_2$- 分级超代数，一个理论不变 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-分级代数意味着存在普通玻色子和三种不同类型的奇异玻色子，其中不同类型的奇异玻色子相互反交换。