To proceed, we assume, as announced above, that the Cartan matrix is invertible and symmetrizable since
these are the only cases encountered in the billiards. Under these assumptions, an invertible, invariant
bilinear form is easily defined on the algebra. We denote by the diagonal elements of
,
Since the bilinear form is nondegenerate on
, one has an isomorphism
defined by
The bilinear form can be uniquely extended from the Cartan subalgebra to the entire algebra by
requiring that it is invariant, i.e., that it fulfills
then unless
. Indeed, one has
and thus
It is proven in [116
Consider the restriction of the bilinear form to the real vector space
obtained by taking the
real span of the simple roots,
When the algebra is infinite-dimensional, the invariant scalar product does not have Euclidean signature.
The spacelike roots are called “real roots”, the non-spacelike ones are called “imaginary roots” [116]. While
the real roots are nondegenerate (i.e., the corresponding eigenspaces, called “root spaces”, are
one-dimensional), this is not so for imaginary roots. In fact, it is a challenge to understand the degeneracy
of imaginary roots for general indefinite Kac–Moody algebras, and, in particular, for Lorentzian
Kac–Moody algebras.
Another characteristic feature of real roots, familiar from standard finite-dimensional Lie algebra theory,
is that if is a (real) root, no multiple of
is a root except
. This is not so for imaginary roots,
where
(or other non-trivial multiples of
) can be a root even if
is. We shall provide explicit
examples below.
Finally, while there are at most two different root lengths in the finite-dimensional case, this is no longer true even for real roots in the case of infinite-dimensional Kac–Moody algebras11. When all the real roots have the same length, one says that the algebra is “simply-laced”. Note that the imaginary roots (if any) do not have the same length, except in the affine case where they all have length squared equal to zero.
The fundamental weights of the Kac–Moody algebra are vectors in the dual space
of the
Cartan subalgebra defined by
The Weyl vector is defined by
From the invariant bilinear form, one can construct a generalized Casimir operator as follows.
We denote the eigenspace associated with by
. This is called the “root space” of
and is
defined as
Let be a basis of
and let
be the basis of
dual to
in the
-metric,
It is proven in [116] that
commutes with all the operators of any restricted representation.
For that reason, it is known as the (generalized) Casimir operator. It is quadratic in the
generators12.
This definition – and, in particular, the presence of the linear term – might seem a bit strange at
first sight. To appreciate it, turn to a finite-dimensional simple Lie algebra. In the above notations, the
usual expression for the quadratic Casimir operator reads
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