Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

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).