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On subalgebras of maximal rank of semisimple Lie algebras. (English) Zbl 0845.17010
Bokut’, L. A. (ed.) et al., Third Siberian school on algebra and analysis. Proceedings of the third Siberian school, Irkutsk State University, Irkutsk, Russia, August 30-September 4, 1989. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 163, 47-60 (1995).
Let $$L$$ be a semisimple Lie algebra over the complex number field, $$L_0 \subset L$$ be a subalgebra of maximal rank. Then the adjoint representation of $$L_0$$ on $$L$$ induces the isotropy representation $$\varepsilon: L_0\to \text{End} (V')$$ of $$L_0$$ on $$V'= L/L_0$$. One states the problem of describing all the cases when the image $$\varepsilon (L_0)$$ in $$\text{End} (V')$$ may be extended to a Lie subalgebra $$G$$ acting irreducibly on $$V'$$. The case when $$V'$$ is the standard representation of classical Lie algebra was already studied. In this paper one supposes that $$V'$$ is not the standard representation of a classical Lie algebra and the complete list of all possibilities for $$(L, L_0, G, V')$$ is determined. The authors consider this problem also in the category of real algebras provided the structure representation is absolutely irreducible.
For the entire collection see [Zbl 0816.00016].
##### MSC:
 17B20 Simple, semisimple, reductive (super)algebras