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
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 .
In , 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 . A similar construction shows that can be regularly embedded in , and that can be regularly embedded in . See  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 . 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 . 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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