Solvable Subalgebras of Classical Lie Algebras

对称性这一日常概念在整个科学界都极为重要。在数学中，它是通过群的概念来研究的，对于这一概念，已经有了高度发达的理论。在许多情况下，特别是那些对物理学感兴趣的人，可以将所谓的李代数与一个对称群联系起来。此相关对象包含与原始组相同的大部分信息，但其形式更易于用于多种用途。具体地说，量子力学的应用通常用李代数而不是对称群来表示。*最常见的对称之一是旋转。平移可能不那么引人注目，但同样为人所熟知。平移是指将某物移动到一个新位置，而不旋转它。组合这两种对称性的群是欧几里德群；从某种意义上说，它描述了普通几何的对称性。相对论最好用一种不同的几何学来表示，有一个对应的群叫做庞加莱群。这两个群分别是研究非相对论和相对论物理学的基础。*他们的李代数，欧几里得和Poincaré代数，包含了许多相关的信息。它们都是数学家所称的半直积代数的例子。这指的是这样一个事实：每个代数都是由两个子代数组合而成的：欧几里德代数中的旋转和平移，以及它们在Poincaré代数中的相对论类似物。*现代物理学的一大挑战是“统一”，即找到一种涵盖所有解释我们物理宇宙不同方面的理论的理论。用数学术语来说，这就提出了将一个代数嵌入另一个代数的问题。粗略地说，这是一个展示描述一种理论的数学结构如何与描述另一种理论的结构相关的问题。这在多大程度上可以或不能以一致的方式完成，对统一这两种理论的可能性具有重要的影响。*本研究项目致力于研究李代数的嵌入问题，包括半直积李代数到其他李代数的嵌入。给出两个李代数，最初的问题是第一个李代数是否可以嵌入第二个李代数。如果可以，那么就有多少种不同的方式可以做到这一点。

The everyday concept of symmetry is extremely important across the sciences. In mathematics, it is studied through the concept of groups, for which there is a highly developed theory. In many cases, especially those of interest in physics, to a symmetry group it is possible to associate what is called a Lie algebra. This related object contains most of the same information as the original group, but in a form that is easier to work with for many purposes. Specifically, applications to quantum mechanics are frequently expressed in terms of Lie algebras rather than symmetry groups. ******One of the most familiar kinds of symmetries is rotation. Perhaps a little less striking, but just as familiar, is translation (moving something to a new position without rotating it). The group that combines these two types of symmetries is the Euclidean group; in a sense it describes the symmetries of ordinary geometry. The theory of relativity can best be expressed in terms of a different kind of geometry, and there is a corresponding group called the Poincaré group. These groups are fundamental in the study of nonrelativistic and relativistic physics, respectively. ******Their Lie algebras, the Euclidean and Poincaré algebras, contain much of the relevant information. They are both examples of what mathematicians call semidirect product algebras. This refers to the fact that each is constructed by combining two subalgebras: the rotations and translations in the Euclidean algebra, and their relativistic analogues in the Poincaré algebra. ******One of the big challenges in modern physics is “unification”, finding a theory which encompasses all the theories that account for different aspects of our physical universe. In mathematical terms, this raises the question of embedding one algebra into another. Roughly speaking, it is a matter of showing how the mathematical structure that describes one theory can be related to the structure that describes another. The extent to which this can or cannot be done in a consistent way has important consequences for the possibility of unifying the two theories. ******This research project is devoted to studying embeddings of Lie algebras, including semidirect product algebras, into other Lie algebras. Given two Lie algebras, there is the initial question whether the first can be embedded in the second. If it can, there is then the question of how many different ways it can be done.**