Lectures on Real Semisimple Lie Algebras and Their Representations (ESI Lectures in Mathematics & Physics) by Arkady L. Onishchik (PDF)

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In 1914, E. Cartan posed the problem of finding all irreducible real linear Lie algebras. Iwahori gave an updated exposition of Cartan’s work in 1959. This theory reduces the classification of irreducible real representations of a real Lie algebra to a description of the so-called self-conjugate irreducible complex representations of this algebra and to the calculation of an invariant of such a representation (with values $+1$ or $-1$) which is called the index. Moreover, these two problems were reduced to the case when the Lie algebra is simple and the highest weight of its irreducible complex representation is fundamental. A complete case-by-case classification for all simple real Lie algebras was given in the tables of Tits (1967). But actually a general solution of these problems is contained in a paper of Karpelevich (1955) that was written in Russian and not widely known. The book begins with a simplified (and somewhat extended and corrected) exposition of the main results of Karpelevich’s paper and relates them to the theory of Cartan-Iwahori. It concludes with some tables, where an involution of the Dynkin diagram that allows for finding self-conjugate representations is described and explicit formulas for the index are given. In a short addendum, written by J. V. Silhan, this involution is interpreted in terms of the Satake diagram. The book is aimed at students in Lie groups, Lie algebras and their representations, as well as researchers in any field where these theories are used. Readers should know the classical theory of complex semisimple Lie algebras and their finite dimensional representation; the main facts are presented without proofs in Section 1. In the remaining sections the exposition is made with detailed proofs, including the correspondence between real forms and involutive automorphisms, the Cartan decompositions and the conjugacy of maximal compact subgroups of the automorphism group. Published by the European Mathematical Society and distributed within the Americas by the American Mathematical Society.

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⭐The main purpose of these notes is to give a self-contained and complete exposition of the representation theory of real semisimple Lie algebras. Although various texts on the topic exist, the originality of this small book is the elegance in the exposition and the presentation of some important facts that are absent in other treatises or only enumerated without further comment. Written by a prestigious expert in Lie theory, the text only demands a standard knowledge in the theory of complex Lie algebras and groups, and constitutes therefore an excellent text as a complement to an advanced course on the classification of complex semisimple Lie algebras and their representation theory.The problem of classifying real simple and semisimple Lie algebras and their representations arises from the geometry of homogeneous spaces, and the first results in this direction were developed by E. Cartan himself in 1914. Using the more standardized algebraic theory and the work of Weyl, the study of real simple Lie algebras and groups was later expanded by various authors in order to develop a self-contained theory in analogy to the complex case. This work accomplishes this objective perfectly, and also pays homage to the important work of the late Fridrikh Izrailevich Karpelevich , who already solved many problems in the representation theory of real simple Lie algebras. However, these papers are unfortunately not widely known in the literature, and various of his results were later rediscovered by other authors. The text is divided into nine sections, which present the main results with detailed proofs and illustrated with examples using the special simple algebra sl(n,C). The choice of this algebra is justified by the role it plays in the characterization of self-dual complex irreducible representations of real forms. For the remaining algebras the reader is led to the references.The first section reviews the classical theory of semisimple complex Lie algebras, and fixes the notation that will be used in later chapters. The main material on compact groups that will be applied in the obtainment of real forms is also briefly presented, such as the theorem of Weyl. As recopilatory chapter, no proofs are given at this stage.The second section deals with the complexification and realification of real and complex Lie algebras, respectively. Two important examples of real forms of complex semisimple Lie algebras are introduced: the real normal form, which can intuitively be interpreted as the algebra obtained by restriction of scalars and the compact form, which will be central for the construction of the remaining non-compact real forms. The first structural results concerning real forms are presented, namely, that real forms of simple complex algebras are simple, while complexification of real simple algebras are either simple or semisimple complex algebras [the insertion of the classical Lorentz algebra would have been welcomed after example 4]. The third section introduces the main tool used in the classification of real forms, the involutive automorphisms of a complex semisimple algebra and its correspondence with the real forms. It follows in particular that the compact form is unique. In order to describe this correspondence, the next section is devoted to various technical results concerning the automorphisms of complex semisimple algebras. Endowed with this machinery, the Cartan decomposition is discussed in detail. The conjugacy theorem of maximal compact subgroups of the adjoint linear group Int(g) is proved. Section 6 is devoted to an important problem which often appears in representation theory: given a homomorphism of complex semisimple Lie algebras f: ĝ →ĥ, which real forms of ĥ contain the image by f of some real form of ĝ? A satisfactory answer to this problem is given by means of the involutive automorphisms corresponding to the real forms. The material of this section follows the original work of F. I. Karpelevich in the beginning fifties . The material previously developed in chapter 3 concerning hermitean vector spaces is applied. Introducing a special class of morphisms, denoted S-homomorphisms , the result is sharpened. The seventh section devotes to the extension problem for irreducible representations for the case of the special linear Lie algebra sl(n,R). Special attention is devoted to the Karpelevich index, and the original formulae for computing this invariant are generalized to arbitrary involutive automorphisms. These results are applied in section 8 to classify explicitly the irreducible real representations in terms of highest weights, following the outline used by Iwahori in 1959. More precisely, real representations divide into two classes depending on the existence or not of an invariant complex structure. The last section, written by J. ?ilhan, presents an alternative classification by means of Satake diagrams, i.e., a generalization of the classical Dynkin diagram based on the introduction of two colors and arrows relating vertices of one color. It is described how to read off the involutions using these diagrams, and a characterization of self-dual complex irreducible representations is obtained. Additional material is presented in tabulated form at the end of this section.Resuming, this book is very welcome reference on real simple Lie algebras, and has the innovation of presenting material that is distributed in many technical papers in a compact and effective way. It should be expected that this work will become a classic on the topic among the specialists in Lie algebras.

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