engleski

Pairs of Lie algebras and their self-normalizing reductive subalgebras

We consider a class P of pairs (g, g_1) of K-Lie algebras $g_1\subseteq g$ satisfying certain "rigidity conditions" ; here K is a field of characteristic 0, and g_1 is reductive. We provide some further evidence that 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)=(sl(n+1), sl(2)) we show that it is in 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 K is algebraically closed, g is semisimple all of whose factors are classical, and s is a principal TDS. Then (g, s) belongs to P. Second ; suppose (g, g_1) is a pair satisfying certain condition (C), and there exists a semisimple $s\subseteq g_1$ such that (g, s) is from P (e.g., s is a principal TDS). Then (g, g_1) is from 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.

Pair of Lie algebras; Semisimple Lie algebra; Reductive subalgebra; Self-normalizing subalgebra; Principal nilpotent element; Principal TDS; Trivial extension

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