Solvable Subalgebras of Classical Lie Algebras

点击翻译按钮获取中文摘要

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.**