
Journal of Lie Theory 19 (2009), No. 4, 735766 Copyright Heldermann Verlag 2009 Pairs of Lie Algebras and their SelfNormalizing Reductive Subalgebras Boris Sirola Dept. of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia sirola@math.hr [Abstractpdf] \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 Killingorthogonal $\p(n)$ 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, selfnormalizing subalgebra, principal nilpotent element, principal TDS, trivial extension. MSC: 17B05, 17B10, 17B20 [ Fulltextpdf (323 KB)] for subscribers only. 