Fine Gradings of Low-Rank Lie Algebras
We introduce the topic of fine gradings of Lie algebras. This concept is important not only for investigating the structural properties of the algebras, but, on top of that, the fine gradings are often used as the starting point for studying graded contractions of the algebras.
We explain how the fine gradings are obtained via several different methods, the basic one using the so-called MAD-groups of the Lie algebras. We explain the different results that the various methods provide for simple complex Lie algebras, non-simple complex Lie algebras, and real forms of both simple and non-simple Lie algebras.
Finally, we focus on the three Lie algebras sl(4,C), sp(4,C), and o(4,C). Inclusions between them are an important tool in our presentation, since they allow to employ all the various methods for finding the fine gradings that we dispose of, and thus perfectly illustrate their effect. Systematic use is made of the faithful representations of the three Lie algebras by 4x4 matrices.