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A. Minchenko Moscow State University Moscow, Russia andrei msu@mail.ru

Triads and short S O3 -subgroups in compact Lie groups
The report is a remark to one of the results of paper [1]. Let G be a compact Lie group. Triad in G is a set of three mutually conjugate involutions (in G), product of which is equal to the identity (thus they commute). A triad is said to be symmetric if any permutation of its elements is realized in G. For example, matrices diag(1, -1, -1), diag(-1, 1, -1), diag(-1, -1, 1) constitute a symmetric triad in SO3 (and any other triad is conjugate to it). A subgroup H G, isomorphic to SO3 , is said to be short SO3 -subgroup if the dimension of any irreducible Ad H -submodule in g does not exceed 5 (i.e. equals 1, 3 or 5). In what follows G is connected simple compact Lie group without center. In [1] all triads and short SO3 -subgroups in G were found up to conjugacy and on the basis of coincidence of the obtained classifications the following theorem was proved. Theorem (Vinb erg). Any triad in G is obtained in a short SO3 -subgroup, which is uniquely determined up to conjugacy. Particularly, all triads in G are symmetric. Vinberg conjectured that there must be some deeper reason for the theorem to be true than just the coincidence of the classifications. The purpose of the report is to show such a reason. The proof is based on the properties of geodesics in symmetric spaces. The fact that sl2 -subalgebra in Lie G(C) is determined, up to conjugacy, by it's semisimple element (length given a priopi) is used as well. References [1] E. B. Vinberg. Short SO3 - structures on simple Lie algebras and the associated quasielliptic planes. Amer. Math. Soc. Transl. 213 (2005), 243--270.

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