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Journal of Lie Theory 19 (2009), No. 4, 735--766
Copyright Heldermann Verlag 2009

Pairs of Lie Algebras and their Self-Normalizing Reductive Subalgebras

Boris Sirola
Dept. of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia


\def\g{{\frak g}} \def\p{{\frak p}} \def\s{{\frak s}} We consider a class $\cal P$ of pairs $(\g,\g_1)$ of {\bf K}-Lie algebras $\g_1\subset\g$ satisfying certain ``rigidity conditions''; here {\bf K} is a field of characteristic $0$, $\g$ is semisimple, and $\g_1$ is reductive. We provide some further evidence that $\cal{P}$ contains a number of nonsymmetric pairs that are worth studying; e.g., in some branching problems, and for the purposes of the geometry of orbits. In particular, for an infinite series $(\g,\g_1) = (\frak{sl}(n+1),\frak{sl}(2))$ we show that it is in $\cal{P}$, and precisely describe a $\g_1$-module structure of the Killing-orthogonal $\p(n)$ of $\g_1$ in $\g$. Using this and the Kostant's philosophy concerning the exponents for (complex) Lie algebras, we obtain two more results. First; suppose $\bf K$ is algebraically closed, $\g$ is semisimple all of whose factors are classical, and $\s$ is a principal TDS. Then $(\g,\s)$ belongs to $\cal{P}$. Second; suppose $(\g,\g_1)$ is a pair satisfying certain technical condition {\bf C}, and there exists a semisimple $\s\subseteq \g_1$ such that $(\g,\s)$ is from $\cal{P}$ (e.g., $\s$ is a principal TDS). Then $(\g,\g_1)$ is from $\cal{P}$ as well. Finally, given a pair $(\g,\g_1)$, we have some useful observations concerning the relationship between the coadjoint orbits corresponding to $\g$ and $\g_1$, respectively.

Keywords: Pair of Lie algebras, semisimple Lie algebra, reductive subalgebra, self-normalizing subalgebra, principal nilpotent element, principal TDS, trivial extension.

MSC: 17B05, 17B10, 17B20

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