Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Abstract

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).