Constructions of Lie algebras and their modules
Let \({\mathfrak g}\) be a central simple Lie algebra over a field of characteristic zero. In these notes the author is interested in constructing all irreducible \({\mathfrak g}\)-modules of finite dimension in terms related to structural descriptions of \({\mathfrak g}\). It is his stated objective to demonstrate the constructions of the irreducible modules in the isotropic cases with non-reduced root systems, and in all anisotropic cases where \({\mathfrak g}\) is neither exceptional nor an exceptional form of \(D_ 4.\) For purposes of motivation and illustration he begins by treating algebras ``of inner type A'' before giving a general discussion of the basic principles of the program to be followed. He then deals with the remaining classical algebras with the generalized even Clifford algebras playing a central part in the construction of the modules. The treatment of exceptional algebras begins by developing constructions of exceptional algebras from quadratic forms in low dimensions and later moves on to constructions of ``super-exceptional'' Lie algebras using the approach of Kantor and Allison. A few additional related topics are briefly discussed in a short final chapter.
isotropic cases with non-reduced root systems
scientific article; zbMATH DE number 192829