标题： 李代数展开方法的一般性质 显示英文标题

标题： General properties of the expansion methods of Lie algebras

摘要： 李代数与群之间的关系研究，特别是从它们中推导出新的代数的问题，在数学和物理学中是一个非常感兴趣的问题，因为从一个已知的李群中找到一个新的李群也意味着可以从一个已知的理论中得到一个新的物理理论。 允许这样做的一个过程称为李代数的展开，并且最近在不同的物理应用中被使用——特别是在引力的规范理论中。 在这里，我们报告了这种方法的进一步发展，以更深入地理解它们在物理理论中的后果。 我们找到了与代数在展开下保持某些性质相关的定理，这些定理可以作为标准，更具体地说，作为判断两个任意李代数是否可以通过某种展开机制相关联的必要条件。 形式方面，如展开代数的卡丹分解，也进行了讨论。 最后，提供了一个有启发性的例子，可以用来明确验证我们所有的理论结果。 显示英文摘要

摘要： The study of the relation between Lie algebras and groups, and especially the derivation of new algebras from them, is a problem of great interest in mathematics and physics, because finding a new Lie group from an already known one also means that a new physical theory can be obtained from a known one. One of the procedures that allow to do so is called expansion of Lie algebras, and has been recently used in different physical applications - particularly in gauge theories of gravity. Here we report on further developments of this method, required to understand in a deeper way their consequences in physical theories. We have found theorems related to the preservation of some properties of the algebras under expansions that can be used as criteria and, more specifically, as necessary conditions to know if two arbitrary Lie algebras can be related by the some expansion mechanism. Formal aspects, such as the Cartan decomposition of the expanded algebras, are also discussed. Finally, an instructive example that allows to check explicitly all our theoretical results is also provided.