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Irreducible orthogonal decompositions in Lie algebras. (Russian) Zbl 0692.17005
Let \({\mathcal L}\) be a simple finite-dimensional complex Lie algebra. A decomposition \({\mathcal L}=\oplus^{n}_{i=1}{\mathcal H}_ i\) of \({\mathcal L}\) into a direct sum of Cartan subalgebras is called an orthogonal decomposition (OD) if these subalgebras \({\mathcal H}_ i\) are mutually orthogonal with respect to the Killing form. Such OD is called irreducible (IOD) if the group \(Aut_{OD}({\mathcal L})\) acts on \({\mathcal L}\) absolutely irreducibly, where \(Aut_{OD}({\mathcal L})=\{\phi |\) \(\phi\in Aut({\mathcal L}) \& \forall i\) \(\exists j\) \(\phi\) (\({\mathcal H}_ i)\subset {\mathcal H}_ j\}\). The main result of the paper is the following theorem which gives a solution of the so called weakened problem of Winnie-the-Pooh [see A. I. Kostrikin, Group theory, Proc. Conf., Singapore 1987, 171-181 (1989; Zbl 0678.17007)].
Theorem. Lie algebras of types \(A_{p-2}\) (p is a prime number, \(p\neq 2^ d-1)\), \(C_ 3\), \(E_ 7\) have no IOD. Lie algebras of types \(A_{p-1}\) (p is a prime number), \(G_ 2\), \(E_ 4\), \(E_ 8\), \(E_ 6\) have IOD, and each of their IOD is standard. The number of Aut(\({\mathcal L})\)-conjugate IOD classes is equal to 1 in the first four cases and is equal to 2 in the last case \(E_ 8\).
Reviewer: G.I.Zhitomirskij

17B20 Simple, semisimple, reductive (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras