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The semisimple subalgebras of exceptional Lie algebras. (English. Russian original) Zbl 1152.17003
Trans. Mosc. Math. Soc. 2006, 225-259 (2006); translation from Tr. Mosk. Mat. O.-va 67, 256-293 (2006).
The author classifies simple subalgebras of exceptional Lie algebras up to conjugacy. In a paper written in 1952 [Transl., II. Ser., Am. Math. Soc. 6, 111–243 (1957); translation from Mat. Sb., Nov. Ser. 30(72), 349–462 (1952; Zbl 0048.01701)], E. B. Dynkin classified the maximal semisimple subalgebras of exceptional Lie algebras up to conjugacy. He also classified the simple subalgebras up to linear conjugacy. If \(L\) is a semisimple Lie algebra two reductive algebras \(S_1\) and \(S_2\) are said to be linearly conjugate if for some finite-dimensional representation of \(L\), these subalgebras become conjugate subalgebras of the corresponding matrix algebra. While conjugate subalgebras are always linearly conjugate, Dynkin himself gave some examples proving that the reciprocal is no longer true. As explained in the work, Dynkin’s idea was to divide the classification task into two steps. In the first, the classification is made up to linear conjugacy. The second should include a study of the partitions on the linear conjugacy classes into conjugacy classes. There are some technical reasons for this approach which the author explains in detail in the paper under review.
This article contains not only the classification of simple subalgebras of exceptional Lie algebras up to conjugacy but also a study of its realizable outer automorphisms. In sections 3 and 4 one can find a description of those semisimple subalgebras of exceptional Lie algebras, whose linear conjugacy class is non-trivially partitioned into conjugacy classes (and the partition is also described). In the final section of the paper, the author finds the normalizers of all connected simple subgroups of rank greater that one in exceptional Lie groups.
As mentioned by the author, the results in this paper are in agreement with results of M. W. Liebeck and G. M. Seitz [Reductive subgroups of exceptional algebraic groups, Mem. Am. Math. Soc. 580 (1996; Zbl 0851.20045)] containing also a classification up to conjugacy of simple subalgebras of exceptional Lie algebras. Finally some inaccuracies contained in the above mentioned work by Dynkin have been corrected in the work under review.

17B25 Exceptional (super)algebras
17B20 Simple, semisimple, reductive (super)algebras
22E10 General properties and structure of complex Lie groups
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[1] A. V. Alekseevskiĭ, Component groups of centralizers of unipotent elements in semisimple algebraic groups, Akad. Nauk Gruzin. SSR Trudy Tbiliss. Mat. Inst. Razmadze 62 (1979), 5 – 27 (Russian, with English summary). Collection of articles on algebra, 2.
[2] È. B. Vinberg, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 488 – 526, 709 (Russian). · Zbl 0363.20035
[3] Семинар по группам Ли и алгебраическим группам, 2нд ед., УРСС, Мосцощ, 1995 (Руссиан, щитх Руссиан суммары). А. Л. Онищик анд Ѐ. Б. Винберг, Лие гроупс анд алгебраиц гроупс, Спрингер Сериес ин Совиет Матхематицс, Спрингер-Верлаг, Берлин, 1990. Транслатед фром тхе Руссиан анд щитх а префаце бы Д. А. Леитес.
[4] Doan Kuin\(^{\prime}\), The Poincaré polynomials of compact homogeneous Riemannian spaces with irreducible stationary group, Trudy Sem. Vektor. Tenzor. Anal. 14 (1968), 33 – 93 (Russian).
[5] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349 – 462 (3 plates) (Russian). · Zbl 0048.01701
[6] V. G. Kac, Automorphisms of finite order of semisimple Lie algebras, Funkcional. Anal. i Priložen. 3 (1969), no. 3, 94 – 96 (Russian).
[7] A. Malcev, On semi-simple subgroups of Lie groups, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 8 (1944), 143 – 174 (Russian, with English summary). · Zbl 0061.04701
[8] I.V. Losev, On invariants of a set of elements of a semisimple Lie algebra, submitted to J. Lie Theory. · Zbl 1239.17003
[9] Martin W. Liebeck and Gary M. Seitz, Reductive subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 121 (1996), no. 580, vi+111. · Zbl 0851.20045
[10] J. McKay, J. Patera, and R. T. Sharp, Second and fourth indices of plethysms, J. Math. Phys. 22 (1981), no. 12, 2770 – 2774. · Zbl 0485.17002
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