The compact and split real Lie algebras constitute the two ends of a string of real forms that can be inferred from a given complex Lie algebra. As announced, this section is devoted to the systematic classification of these various real forms.
If is a real form of , it defines a conjugation on . Indeed we may express any as with and , and the conjugation of with respect to is given by
Conversely, if is a conjugation on , the set of elements of fixed by provides a real form of . Then constitutes the conjugation of with respect to . Thus, on , real forms and conjugations are in one-to-one correspondence. The strategy used to classify and describe the real forms of a given complex simple algebra consists of obtaining all the nonequivalent possible conjugations it admits.
Let be a real form of the complex semi-simple Lie algebra . Consider a compact real form of and the respective conjugations and associated with and . It may or it may not be that and commute. When they do, leaves invariant,
and, similarly, leaves invariant,
Alignment is not automatic. For instance, one can always de-align a compact real form by applying an automorphism to it while keeping unchanged. However, there is a theorem that states that given a real form of the complex Lie algebra , there is always a compact real form associated with it [93, 129]. As this result is central to the classification of real forms, we provide a proof in Appendix B, where we also prove the uniqueness of the Cartan involution.
We shall from now on always consider the compact real form aligned with the real form under study.
A Cartan involution of a real Lie algebra is an involutive automorphism such that the symmetric, bilinear form defined by
A Cartan involution of the real semi-simple Lie algebra yields the direct sum decomposition (called Cartan decomposition)
Define in the algebra by
Conversely, let be a compact real form aligned with and the corresponding conjugation. The restriction of to is a Cartan involution. Indeed, one can decompose as in Equation (6.49), with Equation (6.51) holding since is an involution of . Furthermore, one has also Equation (6.53), which shows that is compact and that is positive definite.
This shows, in view of the result invoked above that an aligned compact real form always exists, that any real form possesses a Cartan involution and a Cartan decomposition. If there are two Cartan involutions, and , defined on a real semi-simple Lie algebra, one can show that they are conjugated by an internal automorphism [93, 129]. It follows that any real semi-simple Lie algebra possesses a “unique” Cartan involution.
On the matrix algebra , the Cartan involution is nothing else than minus the transposition with respect to the scalar product ,
An important consequence of this [93, 129] is that any real semi-simple Lie algebra can be realized as a real matrix Lie algebra, closed under transposition. One can also show [93, 129] that the Cartan decomposition of the Lie algebra of a semi-simple group can be lifted to the group via a diffeomorphism between , where is a closed subgroup with as Lie algebra. It is this subgroup that contains all the topology of .
Let be a real semi-simple Lie algebra. It admits a Cartan involution that allows to split it into eigenspaces of eigenvalue and of eigenvalue . We may choose in a maximal Abelian subalgebra (because the dimension of is finite). The set is a set of symmetric transformations that can be simultaneously diagonalized on . Accordingly we may decompose into a direct sum of eigenspaces labelled by elements of the dual space :
One, obviously non-vanishing, subspace is , which contains . The other nontrivial subspaces define the restricted root spaces of with respect to , of the pair . The that label these subspaces are the restricted roots and their elements are called restricted root vectors. The set of all is called the restricted root system. By construction the different are mutually -orthogonal. The Jacobi identity implies that , while implies that . Thus if is a restricted root, so is .
Let be the centralizer of in . The space is given by
The restricted root space is given by
Note that the multiplicities of the restricted roots might be nontrivial even though the roots are nondegenerate, because distinct roots might yield the same restricted root when restricted to .
Let us denote by the subset of nonzero restricted roots and by the subspace of that they span. One can show [93, 129] that is a root system as defined in Section 4. This root system need not be reduced. As for all root systems, one can choose a way to split the roots into positive and negative ones. Let be the set of positive roots and
The Iwasawa decomposition provides a global factorization of any semi-simple Lie group in terms of closed subgroups. It can be viewed as the generalization of the Gram–Schmidt orthogonalization process.
At the level of the Lie algebra, the Iwasawa decomposition theorem states that
The Iwasawa decomposition of the Lie algebra differs from the Cartan decomposition and is tilted with respect to it, in the sense that is neither in nor in . One of its virtues is that it can be elevated from the Lie algebra to the semi-simple Lie group . Indeed, it can be shown [93, 129] that the map
There is another useful decomposition of in terms of a product of subgroups. Any two generators of are conjugate via internal automorphisms of the compact subgroup . As a consequence writing an element as a product , we may write , which constitutes the so-called decomposition of the group (also sometimes called the Cartan decomposition of the group although it is not the exponention of the Cartan decomposition of the algebra). Here, however, the writing of an element of as product of elements of and is, in general, not unique.
As in the previous sections, is a real form of the complex semi-simple algebra , denotes the conjugation it defines, the conjugation that commutes with , the associated compact aligned real form of and the Cartan involution. It is also useful to introduce the involution of given by the product of the commuting conjugations. We denote it also by since it reduces to the Cartan involution when restricted to . Contrary to the conjugations and , is linear over the complex numbers. Accordingly, if we complexify the Cartan decomposition , to
Let be a -stable Cartan subalgebra of , i.e., a subalgebra such that (i) , and (ii) is a Cartan subalgebra of the complex algebra . One can decompose into compact and noncompact parts,
We have seen that for real Lie algebras, the Cartan subalgebras are not all conjugate to each other; in particular, even though the dimensions of the Cartan subalgebras are all equal to the rank of , the dimensions of the compact and noncompact subalgebras depend on the choice of . For example, for , one may take , in which case , . Or one may take , in which case , .
One says that the -stable Cartan subalgebra is maximally compact if the dimension of its compact part is as large as possible; and that it is maximally noncompact if the dimension of its noncompact part is as large as possible. The -stable Cartan subalgebra used above to introduce restricted roots, where is a maximal Abelian subspace of and a maximal Abelian subspace of its centralizer , is maximally noncompact. If , the Lie algebra constitutes a split real form of . The real rank of is the dimension of its maximally noncompact Cartan subalgebras (which can be shown to be conjugate, as are the maximally compact ones ).
Consider a general -stable Cartan subalgebra , which need not be maximally compact or maximally non compact. A consequence of Equation (6.54) is that the matrices of the real linear transformations are real symmetric for and real antisymmetric for . Accordingly, the eigenvalues of are real (and can be diagonalized over the real numbers) when , while the eigenvalues of are imaginary (and cannot be diagonalized over the real numbers although it can be diagonalized over the complex numbers) when .
Let be a root of , i.e., a non-zero eigenvalue of where is the complexification of the -stable Cartan subalgebra . As the values of the roots acting on a given are the eigenvalues of , we find that the roots are real on and imaginary on . One says that a root is real if it takes real values on , i.e., if it vanishes on . It is imaginary if it takes imaginary values on , i.e., if it vanishes on , and complex otherwise. These notions of “real” and “imaginary” roots should not be confused with the concepts with similar terminology introduced in Section 4 in the context of non-finite-dimensional Kac–Moody algebras.
If is a -stable Cartan subalgebra, its complexification is stable under the involutive authormorphism . One can extend the action of from to by duality. Denoting this transformation by the same symbol , one has
Let be a nonzero root vector associated with the root and consider the vector . One has
Consider now an imaginary root . Then for all and we have , while ; accordingly . Moreover, as the roots are nondegenerate, one has . Writing ascompact, and if it is said to be noncompact.
Suppose that is an imaginary noncompact root. Consider a -root vector . If this root is expressed according to Equation (6.70), then its conjugate, with respect to (the conjugation defined by) , is
The three generators therefore define an subalgebra:
Conversely, if is a real root then . Let be a root vector. Then is also in and hence proportional to . By adjusting the phase of , we may assume that belongs to . At the same time, , also in , is an element of . Evidently, is negative. Introducing (which is in ), we obtain the commutation relations
These two kinds of transformations – called Cayley transformations – allow, starting from a -stable Cartan subalgebra, to transform it into new ones with an increasing number of noncompact dimensions, as long as noncompact imaginary roots remain; or with an increasing number of compact dimensions, as long as real roots remain. Exploring the algebra in this way, we obtain all the Cartan subalgebras up to conjugacy. One can prove that the endpoints are maximally noncompact and maximally compact, respectively.
Theorem: Let be a stable Cartan subalgebra of . Then there are no noncompact imaginary roots if and only if is maximally noncompact, and there are no real roots if and only if is maximally compact .
For a proof of this, note that we have already proven that if there are imaginary noncompact (respectively, real) roots, then is not maximally noncompact (respectively, compact). The converse is demonstrated in .
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