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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

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