# Lectures on Real Semisimple Lie Algebras and Their Representations

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*Arkady L. Onishchik*

A publication of the European Mathematical Society

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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and researchers interested in Lie groups, Lie algebras and their representations and in related fields where these theories are used.