The semisimple subalgebras of exceptional Lie algebras

Minchenko, A.

doi:10.1090/S0077-1554-06-00156-7

The semisimple subalgebras of exceptional Lie algebras
HTML articles powered by AMS MathViewer

by A. N. Minchenko
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2006, 225-259

DOI: https://doi.org/10.1090/S0077-1554-06-00156-7

Published electronically: December 27, 2006

PDF | Request permission

Abstract:

Dynkin classified the maximal semisimple subalgebras of exceptional Lie algebras up to conjugacy, but only classified the simple subalgebras up to the coarser relation of linear conjugacy. In the present paper the simple subalgebras of exceptional Lie algebras are classified up to conjugacy, and their normalizers in the group are found. In a certain sense, this completes the description of the semisimple subalgebras of semisimple Lie algebras. As a by-product we obtain a list of all those semisimple subalgebras of exceptional Lie algebras for which the linear conjugacy class does not coincide with their conjugacy class (in the classical case the corresponding result was known).

References

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. MR 557505
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). MR 0430168
È. B. Vinberg and A. L. Onishchik, Seminar po gruppam Li i algebraicheskim gruppam, 2nd ed., URSS, Moscow, 1995 (Russian, with Russian summary). MR 1403378
Doan Kuin′, The Poincaré polynomials of compact homogeneous Riemannian spaces with irreducible stationary group, Trudy Sem. Vektor. Tenzor. Anal. 14 (1968), 33–93 (Russian). MR 0274665
E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates) (Russian). MR 0047629
V. G. Kac, Automorphisms of finite order of semisimple Lie algebras, Funkcional. Anal. i Priložen. 3 (1969), no. 3, 94–96 (Russian). MR 0251091
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). MR 0011303
I. V. Losev, On invariants of a set of elements of a semisimple Lie algebra, submitted to J. Lie Theory.
Martin W. Liebeck and Gary M. Seitz, Reductive subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 121 (1996), no. 580, vi+111. MR 1329942, DOI 10.1090/memo/0580
W. G. McKay and J. Patera, Tables of dimensions, indices, and branching rules for representations of simple Lie algebras, Lecture Notes in Pure and Applied Mathematics, vol. 69, Marcel Dekker, Inc., New York, 1981. MR 604363