Sentence examples for maximal split in from inspiring English sources

Since S is maximal split in G it is maximal split in the radical of Z G ( S ).

But by Lemma 5.4, S is still maximal split in T over K n and hence over F n because T is split over a Galois extension R ~ n, m / R n for some integer m – a contradiction.

Now it is easy to finish the proof that S is a maximal split torus in G.

But S is a maximal split torus in G. Therefore S = T d ′ = T d and this implies m = m ( S ) = m ( T d ) = m ( T d ′ ).

Then S R ↪ η ( L R ) ⊂ η G R is the maximal split torus in the radical of η ( L I ) R. Since the twist η ( P I ) R ⊗ R F n is a minimal parabolic subgroup of G over F n and η ( L ) I ⊗ R F n is its Levi subgroup it follows that S R ⊗ R F n is a maximal split torus of G over F n ; in particular dim k ( S ) = r.

By Theorem 7.1 m is a MAD subalgebra if and only if T d is a maximal split torus of G, in which case m = m ( T d ).

The following example shows that in general maximal split tori are not necessarily conjugate.

This is in contrast with the concept of maximal split torus which we also need.

We know that the conjugacy of two MAD subalgebras in g is equivalent to the conjugacy of the corresponding maximal split tori.

Since S is central in H it lies inside rad ( H ), hence it is the maximal split torus of rad ( H ). Recall that by Lemma 5.4, S K is still the maximal split torus of rad ( H ) K. If S K is not a maximal split torus of G K, there exists a split torus S ′ of H K such that S ′ is not a subgroup of rad ( H K ).

Every maximal split torus of G is generically maximal split.