# On root systems and an infinitesimal classification of irreducible symmetric spaces

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## Abstract

Jou rn al of M athem atics, O saka City University, V ol. 13, N o. I . On root systems and an infinitesimal classification of irreducible symmetric spaces by Shoro A rak i {Received May 8, 1962) Introduction. The classification of real simple Lie algebras was given first by E.Cartan [2] in 1914. Though his first classification lacked in general theorems, Cartan himself [5] estabhshed in 1929 a general theorem suitable to sim'pHfy the classification. Then Gantmacher [6] in 1939 gave a simplified classification depending on Cartan’s general theorem by making use of his theory on canonical representation of automor phisms of complex semi-simple Lie groups. In his earlier papers [5, 3] Cartan established a priori a one-one correspondence between non-compact real simple Lie algebras and irreducible infinitesimal symmetric spaces (compact or non-compact), where “infinitesimal” means locally isomorphic classes. Hence the infinitesimal classification of irreducible symmetric spaces is the same thing as the classification of non-compact real simple Lie algebras. Let g be a real semi-simple Lie algebra and f a maximal compact subalgebra of g. Then we have a Cartan decomposition g = t + p , where p is the orthogonal complement of f with respect to the Killing form. In the classical theories of classification of real simple Lie algebras due to E.Cartan and Gantmacher, one used a Cartan subalgebra ofg whose torus part f) ^ is maximal abelian in f, whereas certain geometric objects (such as roots, geodesics etc.) of symmetric spaces are related to a Cartan subalgebra ^ of g whose vector part fl is maximal abelian in p (cf., Cartan [4], Bott-Samelson [I] and Satake [7]). The two types of Cartan subalgebras mentioned above are not the same and even non conjugate to each other in general. So it seems preferable to the author to have a classification theory by making use of the latter Cartan subalgebra, so as to connect it more closely with the theory of roots of symmetric

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