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 ,

First, one defines an invertible bilinear form in the dual of the Cartan subalgebra. This is done by simply setting for the simple roots. It follows from that and thus the Cartan matrix can be written as It is customary to fix the normalization of so that the longest roots have . As we shall now see, the definition (4.29) leads to an invariant bilinear form on the Kac–Moody algebra.Since the bilinear form is nondegenerate on , one has an isomorphism defined by

This isomorphism induces a bilinear form on the Cartan subalgebra, also denoted by . The inverse isomorphism is denoted by and is such that Since the Cartan elements obey it is clear from the definitions that and thus alsoThe 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

For instance, for the ’s and ’s one finds and similarly In the same way we have and thus Quite generally, if and are root vectors corresponding respectively to the roots and ,then unless . Indeed, one has

and thus

It is proven in [116] 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,

This defines a scalar product with a definite signature. As we have mentioned, the signature is Euclidean if and only if the algebra is finite-dimensional [116]. 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” [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
algebras^{11}.
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

This impliesThe Weyl vector is defined by

and is thus equal to

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

A representation of the Kac–Moody algebra is called restricted if for every vector of the representation subspace , one has for all but a finite number of positive roots .Let be a basis of and let be the basis of dual to in the -metric,

Similarly, let be a basis of and the dual basis of with respect to the bilinear form , We set where is the Weyl vector. Recall from Section 4.6.1 that is an isomorphism from to , so, since , the expression belongs to as required. When acting on any vector of a restricted representation, is well-defined since only a finite number of terms are different from zero. 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
generators^{12}.

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

(without degeneracy index since the roots are nondegenerate in the finite-dimensional case). Here, is the Killing metric and a basis of the Lie algebra. This expression is not “normal-ordered” because there are, in the last term, lowering operators standing on the right. We thus replace the last term by Using the fact that in a finite-dimensional Lie algebra, , (see, e.g., [85]) 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.http://www.livingreviews.org/lrr-2008-1 |
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