By Arkady L. Onishchik

In 1914, E. Cartan posed the matter of discovering all irreducible actual linear Lie algebras. Iwahori gave an up to date exposition of Cartan's paintings in 1959. This conception reduces the category of irreducible genuine representations of a true Lie algebra to an outline of the so-called self-conjugate irreducible complicated representations of this algebra and to the calculation of an invariant of any such illustration (with values $+1$ or $-1$) called the index. additionally, those difficulties have been decreased to the case whilst the Lie algebra is straightforward and the top weight of its irreducible complicated illustration is key. a whole case-by-case type for all basic actual Lie algebras used to be given within the tables of knockers (1967). yet truly a basic answer of those difficulties is contained in a paper of Karpelevich (1955) that used to be written in Russian and never well known.

The publication starts with a simplified (and a bit prolonged and corrected) exposition of the most result of Karpelevich's paper and relates them to the speculation of Cartan-Iwahori. It concludes with a few tables, the place an involution of the Dynkin diagram that permits for locating self-conjugate representations is defined and specific formulation for the index are given. In a quick addendum, written via J. V. Silhan, this involution is interpreted when it comes to the Satake diagram.

The ebook is aimed toward scholars in Lie teams, Lie algebras and their representations, in addition to researchers in any box the place those theories are used. Readers should still be aware of the classical conception of advanced semisimple Lie algebras and their finite dimensional illustration; the most evidence are provided with no proofs in part 1. within the final sections the exposition is made with unique proofs, together with the correspondence among actual kinds and involutive automorphisms, the Cartan decompositions and the conjugacy of maximal compact subgroups of the automorphism staff.

Published via the eu Mathematical Society and allotted in the Americas through the yank Mathematical Society.

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It follows that C = λ2 C, whence λ = ±1. Thus, we get two different cases. I. If λ = 1, then we have C = C. The reduction theory of symmetric bilinear forms implies that C = U U , where the matrix U can be chosen from SLn (C). Then θ(X) = −U U X (U )−1 U −1 = (Ad U )ω(Ad U )−1 (X) , X ∈ sln (C) . Thus, θ = (Ad U )ω(Ad U )−1 is conjugate to ω by an inner automorphism of g. II. If λ = −1, then we have C = −C and n = 2m. 5). It follows that θ = (Ad U )(Ad Sm )ω(Ad U )−1 is conjugate to (Ad Sm )ω by an inner automorphism of g.

Now, Proposition 1 (ii), implies that ad z ∈ S(g0 ), and hence exp(ad z) ∈ P for any z ∈ p. Theorem 2. The mappings µ : K × P → Aut g0 and µ(id × exp ad) : K × p → Aut g0 are bianalytic and induce bianalytic maps K ◦ × P → Int g0 and K ◦ × p → Int g0 . We have K ◦ = K ∩ Int g0 = {α ∈ Int g0 | θαθ = α} . Proof. To prove our claim concerning Aut g0 , it suffices to verify that for any a ∈ Aut g0 the operators s ∈ S(g0 ) and p ∈ P(g0 ) expressed by (10) belong to ad p and Aut g0 , respectively. By Proposition 1 (i), we see that a∗ ∈ Aut g0 , and therefore p2 ∈ Aut g0 .

Then θt ∈ Aut g for all t ∈ R. If g is semisimple, then θt ∈ Int g for all t ∈ R. k Proof. Let g = i=1 gλi be the eigenspace decomposition for θ, where λi > 0 are the eigenvalues. If x ∈ gλi , y ∈ gλj , then θ[x, y] = [θx, θy] = [λi x, λj y] = λi λj [x, y] , whence [x, y] ∈ gλi λj . Therefore θt [x, y] = (λi λj )t [x, y] = [λti x, λtj y] = [θt x, θt y] for any t ∈ R. It follows that θt ∈ Aut g. 7). A compact real structure in a complex Lie algebra is a real structure, whose corresponding real form is compact.

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