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
Indeed any element 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 [129
]).
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
as
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 [129
].
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 [129
].
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