We have shown in the previous sections that Weyl groups of Lorentzian Kac–Moody algebras naturally
emerge when analysing gravity in the extreme BKL regime. This has led to the conjecture that the
corresponding Kac–Moody algebra is in fact a symmetry of the theory (most probably enlarged with new
fields) [46]. The idea is that the BKL analysis is only the “revelator” of that huge symmetry, which would
exist independently of that limit, without making the BKL truncations. Thus, if this conjecture is true,
there should be a way to rewrite the gravity Lagrangians in such a way that the Kac–Moody symmetry is
manifest. This conjecture itself was made previously (in this form or in similar ones) by other authors on
the basis of different considerations [113
, 139, 167
]. To explore this conjecture, it is desirable to have a
concrete method of dealing with the infinite-dimensional structure of a Lorentzian Kac–Moody algebra
.
In this section we present such a method.
The method by which we shall deal with the infinite-dimensional structure of a Lorentzian Kac–Moody
algebra is based on a certain gradation of
into finite-dimensional subspaces
. More precisely, we
will define a so-called level decomposition of the adjoint representation of
such that each level
corresponds to a finite number of representations of a finite regular subalgebra
of
.
Generically the decomposition will take the form of the adjoint representation of
plus a
(possibly infinite) number of additional representations of
. This type of expansion of
will prove to be very useful when considering sigma models invariant under
for which we
may use the level expansion to consistently truncate the theory to any finite level
(see
Section 9).
We begin by illustrating these ideas for the finite-dimensional Lie algebra after which we
generalize the procedure to the indefinite case in Sections 8.2, 8.3 and 8.4.
http://www.livingreviews.org/lrr-2008-1 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |