Structure of Semisimple Lie Groups
Chapter I presents a brief resume, with occasional indications of proofs, of the theory of semisimple Lie groups up to (but not including) Cartan’s highest weight theory for finite-dimensional representations and the theory of parabolic subgroups. We start with some basic notions of linear algebra (Section 1) and do the representations of sl(2) which have a highest or lowest weight vector (Section 2). We then discuss some basic facts about semisimple (Section 3) and reductive (Section 4) groups. After that, we look at root systems and the Weyl group (Section 5), Weyl bases and real forms (Section 6), Dynkin diagrams and classification (Section 7), and have a brief glimpse of the structure of real semisimple groups (Section 8). All this can be viewed as a sort of study guide to Varadarajan’s book, through Chapter 4, Section 5.
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