Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

摘要

Let $mathfrak g$ be a semisimple Lie algebra over a field $mathbb K$ , $text{char}left( mathbb{K} right)=0$ , and $mathfrak g_1$ a subalgebra reductive in $mathfrak g$ . Suppose that the restriction of the Killing form B of $mathfrak g$ to $mathfrak g_1 times mathfrak g_1$ is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra $mathfrak h_1$ of $mathfrak g_1$ there is a unique Cartan subalgebra $mathfrak h$ of $mathfrak g$ containing $mathfrak h_1$ ; ( 2) $mathfrak g_1$ is self-normalizing in $mathfrak g$ ; ( 3) The B-orthogonal $mathfrak p$ of $mathfrak g_1$ in $mathfrak g$ is simple as a $mathfrak g_1$ -module for the adjoint representation. We give some answers to this natural question: For which pairs $(mathfrak g,mathfrak g_1)$ do ( 1), ( 2) or ( 3) hold? We also study how $mathfrak p$ in general decomposes as a $mathfrak g_1$ -module, and when $mathfrak g_1$ is a maximal subalgebra of $mathfrak g$ . In particular suppose $(mathfrak g,sigma )$ is a pair with $mathfrak g$ as above and σ its automorphism of order m. Assume that $mathbb K$ contains a primitive m-th root of unity. Define $mathfrak g_1:=mathfrak g^{sigma}$ , the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) $(mathfrak g,mathfrak g_1)$ satisfies ( 1); (b) For m prime, $(mathfrak g,mathfrak g_1)$ satisfies ( 2).