In the previous example, we performed the decomposition of the roots (and the ensuing decomposition
of the algebra) with respect to one of the simple roots which then defined the level. In general, one may
consider a similar decomposition of the roots of a rank Kac–Moody algebra with respect to an arbitrary
number
of the simple roots and then the level
is generalized to the “multilevel”
.
We consider a Kac–Moody algebra of rank
and we let
be a finite regular rank
subalgebra of
whose Dynkin diagram is obtained by deleting a set of nodes
from the Dynkin diagram of
.
Let be a root of
,
We note for further reference that the following structure is inherited from the gradation:
This implies that for At level zero, , the representation of the subalgebra
in the subspace
contains
the adjoint representation of
, just as in the case of
discussed in Section 8.1. All positive and
negative roots of
are relevant. Level zero contains in addition
singlets for each of the Cartan
generator associated to the set
.
Whenever one of the ’s is positive, all the other ones must be non-negative for the subspace
to
be nontrivial and only positive roots appear at that value of the multilevel.
Let be the module of a representation
of
and
be one of the weights occurring in
the representation. We define the action of
in the representation
on
as
The dual space may be viewed as the
-dimensional subspace
of
spanned by the simple
roots
,
. The metric induced on that subspace is positive definite since
is finite-dimensional.
This implies, since we assume that the metric on
is nondegenerate, that
can be decomposed as
the direct sum
It follows that one can identify the weight with the orthogonal projection
of
on
. This is true, in particular, for the fundamental weights
. The fundamental weights
project on
for
and project on the fundamental weights
of the subalgebra
for
. These are also denoted
. For a general weight, one has
There is an interesting relationship between root multiplicities in the Kac–Moody algebra and weight
multiplicites of the corresponding
-weights, which we will explore here.
For finite Lie algebras, the roots always come with multiplicity one. This is in fact true also for the real
roots of indefinite Kac–Moody algebras. However, as pointed out in Section 4, the imaginary
roots can have arbitrarily large multiplicity. This must therefore be taken into account in the
sum (8.13).
Let be a root of
. There are two important ingredients:
It follows that the root multiplicity of is given as a sum over its multiplicities as a weight in the various
representations
at level
. Some representations can appear more than once at
each level, and it is therefore convenient to introduce a new measure of multiplicity, called the outer
multiplicity
, which counts the number of times each representation
appears at level
. So,
for each representation at level
we must count the individual weight multiplicities in that representation
and also the number of times this representation occurs. The total multiplicity of
can then be written
as
The subspaces can now be written explicitly as
Finally, we mention that the multiplicity of a root
can be computed recursively
using the Peterson recursion relation, defined as [116
]
|
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 |