The Weyl group enjoys a number of interesting properties [116]:
This close relationship between Coxeter groups and Kac–Moody algebras is the reason for denoting both
with the same notation (for instance, denotes at the same time the Coxeter group with Coxeter graph
of type
and the Kac–Moody algebra with Dynkin diagram
).
Note that different Kac–Moody algebras may have the same Weyl group. This is in fact already
true for finite-dimensional Lie algebras, where dual algebras (obtained by reversing the arrows
in the Dynkin diagram) have the same Weyl group. This property can be seen from the fact
that the Coxeter exponents are related to the duality-invariant product . But, on top
of this, one sees that whenever the product
exceeds four, which occurs only in the
infinite-dimensional case, the Coxeter exponent
is equal to infinity, independently of the
exact value of
. Information is thus clearly lost. For example, the Cartan matrices
Because the Weyl groups are (crystallographic) Coxeter groups, we can use the theory of Coxeter groups to analyze them. In the Kac–Moody context, the fundamental region is called “the fundamental Weyl chamber”.
We also note that by (standard vector space) duality, one can define the action of the Weyl group in the
Cartan subalgebra , such that
Finally, we leave it to the reader to verify that when the products are all
, then the
geometric action of the Coxeter group considered in Section 3.2.4 and the geometric action of
the Weyl group considered here coincide. The (real) roots and the fundamental weights differ
only in the normalization and, once this is taken into account, the metrics coincide. This is
not the case when some products
exceed 4. It should be also pointed out that the
imaginary roots of the Kac–Moody algebras do not have immediate analogs on the Coxeter
side.
As the first (respectively, second) Cartan matrix defines the Lie algebra (respectively
)
introduced below in Section 4.9, we also write it as
(respectively,
).
We denote the associated sets of simple roots by
and
, respectively. In
both cases, the Coxeter exponents are
,
,
and the metric
of the
geometric Coxeter construction is
We associate the simple roots with the geometric realisation of the Coxeter group
defined by the matrix
. These roots may a priori differ by normalizations from
the simple roots of the Kac–Moody algebras described by the Cartan matrices
and
.
Choosing the longest Kac–Moody roots to have squared length equal to two yields the scalar products
Recall now from Section 3 that the fundamental reflections have the following geometric
realisation
We now want to compare this geometric realisation of with the action of the Weyl groups of
and
on the corresponding simple roots
and
. According to Equation (4.56
), the Weyl
group
acts as follows on the roots
while the Weyl group acts as
We see that the reflections coincide, ,
,
, as
well as the scalar products, provided that we set
,
,
and
. The Coxeter group
generated by the reflections thus preserves the lattices
It follows, of course, that the Weyl groups of the Kac–Moody algebras and
are the
same,
and its symmetrization
The Weyl group of the corresponding Kac–Moody algebra is isomorphic to the Coxeter
group
above since, according to the rules, the Coxeter exponents are identical. But the action is
now
and cannot be made to coincide with the previous action by rescalings of the ’s. One can easily
convince oneself of the inequivalence by computing the eigenvalues of the matrices
,
and
with respect to
.
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