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 that the invariance condition on the bilinear form defines it indeed consistently and that it is nondegenerate. Furthermore, one finds the relations
Consider the restriction of the bilinear form to the real vector space obtained by taking the real span of the simple roots,. In that case, all roots – and not just the simple ones – are spacelike, i.e., such that .
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” . 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  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) one sees that the Casimir operator can be rewritten in “normal ordered” form as in Equation (4.52). The advantage of the normal-ordered form is that it makes sense also for infinite-dimensional Kac–Moody algebras in the case of restricted representations.
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