LEADER 03364nam 22004095a 450 001 9910151938803321 005 20091109150325.0 010 $a3-03719-502-9 024 70$a10.4171/002 035 $a(CKB)3710000000953787 035 $a(CH-001817-3)18-091109 035 $a(PPN)178154989 035 $a(EXLCZ)993710000000953787 100 $a20091109j20040228 fy 0 101 0 $aeng 135 $aurnn|mmmmamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aLectures on Real Semisimple Lie Algebras and Their Representations$b[electronic resource] /$fArkady L. Onishchik 210 3 $aZuerich, Switzerland $cEuropean Mathematical Society Publishing House$d2004 215 $a1 online resource (95 pages) 225 0 $aESI Lectures in Mathematics and Physics (ESI) 330 $aIn 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 con... 606 $aAlgebra$2bicssc 606 $aNonassociative rings and algebras$2msc 606 $aTopological groups, Lie groups$2msc 615 07$aAlgebra 615 07$aNonassociative rings and algebras 615 07$aTopological groups, Lie groups 686 $a17-xx$a22-xx$2msc 700 $aOnishchik$b Arkady L.$0535921 801 0$bch0018173 906 $aBOOK 912 $a9910151938803321 996 $aLectures on real semisimple Lie algebras and their representations$91106383 997 $aUNINA