Lectures on real semisimple Lie algebras and their representations
The author's primary goal is to provide an introduction to real semisimple Lie algebras and their representations. This is a welcome addition to the literature, given that this material, in addition to being interesting in its own right, is prerequisite for understanding the orbit structure of homogeneous spaces and the associated realizations of real semisimple Lie group representations. The book begins with a brief review of certain aspects of elementary Lie theory, and then proceeds to investigate real forms of a complex Lie algebra \({\mathfrak g}\) and the correspondence between real forms of \({\mathfrak g}\) and involutions of \({\mathfrak g} \). Later, after a discussion of automorphisms of complex Lie algebras and Cartan decompositions, the author considers the following problem: Given a finite dimensional complex representation \(\rho:{\mathfrak g} \rightarrow {\mathfrak sl} (V)\) and an involution \(\theta\in \Aut\,{\mathfrak g}\), which involutions in \( \Aut\,{\mathfrak sl}(V)\) are extensions of \(\theta\) by \(\rho\)? The solution boils down to the problem of determining inclusions of images of real forms of \({\mathfrak g}\) under \(\rho\) in real forms of \({\mathfrak sl}(V)\). This material on inclusion pays homage to the early work of Karpelevich, who used these types of methods in problems related to the classification real representations of real Lie algebras. The book then finishes with a classification of the irreducible real representations of real semisimple Lie algebras, including a section of Satake diagrams.
scientific article; zbMATH DE number 2052526