Let be a Kac–Moody algebra, and let
be a subalgebra of
with triangular decomposition
. We assume that
is canonically embedded in
, i.e., that the Cartan subalgebra
of
is a subalgebra of the Cartan subalgebra
of
,
, so that
. We shall say
that
is regularly embedded in
(and call it a “regular subalgebra”) if and only if two
conditions are fulfilled: (i) The root generators of
are root generators of
, and (ii) the
simple roots of
are real roots of
. It follows that the Weyl group of
is a subgroup of
the Weyl group of
and that the root lattice of
is a sublattice of the root lattice of
.
The second condition is automatic in the finite-dimensional case where there are only real roots. It must
be separately imposed in the general case. Consider for instance the rank 2 Kac–Moody algebra with
Cartan matrix
Let
It is easy to verify that
We shall describe some regular subalgebras of . The Dynkin diagram of
is displayed in
Figure 21
.
A first, simple, example of a regular embedding is the embedding of in
which will be used to
define the level when trying to reformulate eleven-dimensional supergravity as a nonlinear sigma model.
This is not a maximal embedding since one can find a proper subalgebra
of
that contains
. One may take for
the Kac–Moody subalgebra of
generated by the operators
at levels
and
, which is a subalgebra of the algebra containing all operators of even
level15.
It is regularly embedded in
. Its Dynkin diagram is shown in Figure 22
.
In terms of the simple roots of , the simple roots of
are
through
and
. The algebra
is Lorentzian but not hyperbolic. It can be
identified with the “very extended” algebra
[86].
In [67], Dynkin has given a method for finding all maximal regular subalgebras of finite-dimensional simple
Lie algebras. The method is based on using the highest root and is not generalizable as such to
general Kac–Moody algebras for which there is no highest root. Nevertherless, it is useful for
constructing regular embeddings of overextensions of finite-dimensional simple Lie algebras. We
illustrate this point in the case of and its overextension
. In the notation of
Figure 21
, the simple roots of
(which is regularly embedded in
) are
and
.
Applying Dynkin’s procedure to , one easily finds that
can be regularly embedded in
. The simple roots of
are
,
and
, where
This is done as follows. It is reasonable to guess that the searched-for Weyl element that maps the “old”
on the “new”
is some product of the Weyl reflections in the four
-roots orthogonal to the
simple roots
,
,
,
and
, expected to be shared (as simple roots) by
,
the old
and the new
– and therefore to be invariant under the searched-for Weyl
element. This guess turns out to be correct: Under the action of the product of the commuting
-Weyl reflections in the
-roots
and
, the set of
-roots
is
mapped on the equivalent set of positive roots
, where
The embedding just described is in fact relevant to string theory and has been discussed from various
points of view in previous papers [125, 23
]. By dimensional reduction of the bosonic sector of
eleven-dimensional supergravity on a circle, one gets, after dropping the Kaluza–Klein vector and the
3-form, the bosonic sector of pure
ten-dimensional supergravity. The simple roots of
are
the symmetry walls and the electric and magnetic walls of the 2-form and coincide with the positive roots
given above [45
]. A similar construction shows that
can be regularly embedded in
, and that
can be regularly embedded in
. See [106] for a recent discussion of
in the
context of Type I supergravity.
As we have just seen, the raising operators of might be raising or lowering operators of
. We
shall consider here only the case when the positive (respectively, negative) root generators of
are also positive (respectively, negative) root generators of
, so that
and
(“positive regular embeddings”). This will always be assumed from now
on.
In the finite-dimensional case, there is a useful criterion to determine regular algebras from subsets of
roots. This criterion, which does not use the highest root, has been generalized to Kac–Moody algebras
in [76]. It covers also non-maximal regular subalgebras and goes as follows:
Theorem: Let be the set of positive real roots of a Kac–Moody algebra
. Let
be
chosen such that none of the differences
is a root of
. Assume furthermore that the
’s are
such that the matrix
has non-vanishing determinant. For each
,
choose non-zero root vectors
and
in the one-dimensional root spaces corresponding to the
positive real roots
and the negative real roots
, respectively, and let
be
the corresponding element in the Cartan subalgebra of
. Then, the (regular) subalgebra of
generated by
,
, is a Kac–Moody algebra with Cartan matrix
.
Proof: The proof of this theorem is given in [76]. Note that the Cartan integers are indeed
integers (because the
’s are positive real roots), which are non-positive (because
is not a root),
so that
is a Cartan matrix.
As we have mentioned above, it is convenient to universally normalize the Killing form of Kac–Moody
algebras in such a way that the long real roots have always the same squared length, conveniently taken
equal to two. It is then easily seen that the Killing form of any regular Kac–Moody subalgebra of
coincides with the invariant form induced from the Killing form of
through the embedding since
is “simply laced”. This property does not hold for non-regular embeddings as the example given in
Section 4.1 shows: The subalgebra
considered there has an induced form equal to minus the standard
Killing form.
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