ProofTake any abelian subalgebra H of the Lie algebra G, and decompose G into adirect sum of generalized eigenspace of H (acting on G by the adjoint represen-tation). The eigenvalues are elements of the dual of H. If Gλis the generalizedeigenspace for some eigenvalue λ, then [Gλ, Gµ] ⊆ Gλ+µ. In particular G0is aself-normalizing subalgebra of G containing H. If in addition all elements of Hare semisimple, then H lies in the center of G0as generalized eigenvectors (witheigenvalue 0) are honest eigenvectors. For semisimple Lie algebras we will later see that a maximal toral subalgebrais its own normalizer (so is a Cartan subalgebra). In general this is not true:Exercise 137 Show that a subalgebra of a nilpotent Lie algebra is toral if andonly if it is contained in the center. So the center is the unique maximal toralsubalgebra, and its normalizer is the whole algebra.This exercise is a bit misleading. There is a subtle problem in that the definitinoof a maximal toral subgroup of an algebraic group does not quite correspond tomaximal toral subalgebras of the Lie algebra. This is because a toral subgroupof an algebraic group has elements that are semisimple. In (say) the group ofunipotent upper triangular matrices, the only semisimple element is 1, so themaximal toral subgroup is trivial. However the maximal toral subalgebra ofits Lie algebra is the center which is not trivial. This is related to the factthat it is ambiguous whether elements of the center of a Lie algebra or groupshould be thought of as semisimple or unipotent/nilpotent. For example, in theHeisenberg algebra, the center looks nilpotent in finite dimensional (algebraic)representations, but looks semisimple in its standard infinite dimensional repre-sentation. The Heisenberg algebra is trying hard to be semisimple in some sense;in fact it can be thought of as a sort of degeneration of a semisimple algebra.For semisimple lie algebras or groups this problem does not arise: “maximaltoral” means the same whether one defines it algebraically or analytically.Exercise 138 Show that sl2(R) has two maximal toral subalgebras that arenot conjugate under any automorphism. (Take one to correspond to diagonalmatrices, and the other to correspond to a compact group of rotations.)Theorem 139 If a finite dimensional complex Lie algebra is semisimple, thenthe normalizers of the maximal toral subalgebras are abelianProof Suppose H is a maximal toral subalgebra, and G0its normalizer, so thatG0is nilpotent. Since G0is solvable it can be put into upper triangular form, sothe Killing form restricted to G0has [G0, G0] in its kernel. On the other hand,any invariant bilinear form vanishes on (u, v) if u and v have eigenvalues thatdo not sum to 0, so G0is orrthogonal to all other eigenspaces of H. So [G0, G0]is in the kernel of the Killing form. By Cartan’s criterion, this implies that itvanishes, so G0is abelian. Remark 140 There is an analogue of Cartan subgroups for finite groups. Asubgroup of a finite group is called a Carter subgroup (not a misprint: these arenamed after Roger Carter) if it is nilpotent and self-normalizing. Any solvablefinite group contains Carter subgroups, and any two Carter subgroups of a finite53group are conjugate. However anyone with plans to classify the finite simplegroups by copying the use of Cartan subgroups in the classification of simpleLie groups should take note of the following exercise:Exercise 141 Show that the s imple group A5of order 60 does not have anyCarter subgroups.The analogues of Cartan subgroups for compact Lie groups are maximaltori. In fact these are the subgroups associated to Cartan subalgebras of theLie algebra. Every element of a compact connected Lie group is contained in amaximal torus, and the maximal tori are all conjugate.Warning 142 In a compact connected Lie group, maximal tori are maximalabelian subgroups, but the converse is false in general: maximal abelian sub-groups of a compact connected Lie group are not necessarily maximal tori. Thisis a common mistake. In particular, although every element is contained in atorus, it need not be true that every abelian subgroup is contained in a torus.Exercise 143 Show that the subgroup of diagonal matrices of SOn(R) for n ≥3 is a maximal abelian subgroup but is not contained in any torus.12 Unitary and general linear groupsThe fundamental example of a Lie group is the general linear group GLn(R).There are several closely related variations of this:• The complex general linear group GLn(C)• The unitary group Un• The special linear groups or special unitary groups, where one restricts tomatrices of determinant 1• The projective linear groups, where one quotients out by the center (di-agonal matrices)Exercise 144 Show that the complex Lie algebras gln(R) ⊗ C, gln(C), andun(R) ⊗ C are all isomorphic. We say that gln(R) and un(R) are real forms ofgln(C).The general linear group has a rather obvious representation on n-dimensionalspace. Therefore it also acts on the 1-dimensional subspaces of this, in otherwords n − 1-dimensional projective space. The center acts trivially, so we getan action of the projective general linear group P GLn(R) on Pn−1. There isnothing special about 1-dimensional subspaces: the general linear group alsoacts on the Grassmannian G(m, n − m) of m-dimensional subspaces of Rn. Thesubgroup fixing one such subspace is the subgroup of block matrices (∗ ∗0 ∗), sothe Grassmannian is a quotient of these two group.Exercise 145 Show that the Grassmannian is compact. (This also follows fromthe Iwasawa decomposition below).54More generally still, we can let the general linear group act on the flag mani-folds, consisting of chains of subspaces 0 ⊂ V1⊂ V2· · · , where the subspaceshave given dimensions. The extreme case is when Vihas dimension i, in whichcase the subgroup fixing a flag is the Borel subgroup of upper triangular ma-trices. In general the subgroups fixing flags are called parabolic subgroups; theorresponding quotient spaces are projective varieties.The Iwasawa decomposition for the general linear group is G = GLn(R) =KAN where K = On(R) is a maximal compact subgroup, A is the abeliansubgroup of diagonal matrices with positive coefficients, and N is the unipotentsubgroup of unipotent upper triangular matrices. For the general linear group,the Iwasawa decomposition is essentially the same as the Gram-Schmidt processfor turning a base into an orthonormal base. This works as