# Book Details

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Contents | Introduction | MARC record | Metadata XML | e-Book PDF (1195 KB)*Arkady L. Onishchik (Yaroslavl State University, Russia)*

#### Lectures on Real Semisimple Lie Algebras and Their Representations

ISBN print 978-3-03719-002-9, ISBN online 978-3-03719-502-4DOI 10.4171/002

February 2004, 95 pages, softcover, 17 x 24 cm.

24.00 Euro

In 1914, E. Cartan posed the problem to find all irreducible real
linear Lie algebras. An updated exposition of his work was given by
Iwahori (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 (without proof) in the tables of Tits (1967). But
actually a general solution of these problems is contained in a paper
of Karpelevich (1955) (written in Russian and not widely known), where
inclusions between real forms induced by a complex representation were
studied.

We begin with a simplified (and somewhat extended and corrected)
exposition of the main part of this paper and relate it to the theory
of Cartan-Iwahori. We conclude with some tables, where an involution of
the Dynkin diagram which allows us to find 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. The reader is supposed to 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.

#### Further Information

Review in Zentralblatt MATH 1125.53036