Go to previous page Go up Go to next page

3.6 Crystallographic Coxeter groups

Among the Coxeter groups, only those that are crystallographic correspond to Weyl groups of Kac–Moody algebras. Therefore we now introduce this important concept. By definition, a Coxeter group is crystallographic if it stabilizes a lattice in V. This lattice need not be the lattice generated by the αi’s. As discussed in [107], a Coxeter group is crystallographic if and only if two conditions are satisfied: (i) The integers mij (i ⁄= j) are restricted to be in the set {2, 3,4,6,∞ }, and (ii) for any closed circuit in the Coxeter graph of ℭ, the number of edges labelled 4 or 6 is even.

Given a crystallographic Coxeter group, it is easy to exhibit a lattice L stabilized by it. We can construct a basis for that lattice as follows. The basis vectors μ i of the lattice are multiples of the original simple roots, μi = ciαi for some scalars ci which we determine by applying the following rules:

One easily verifies that σi(μj) = μj − dijμi for some integers dij. Hence L is indeed stabilized. The integers d ij are equal to 2 B(μi,μj)- B(μi,μi).

The rules are consistent as can be seen by starting from an arbitrary node, say α1, for which one takes c1 = 1. One then proceeds to the next nodes in the (connected) Coxeter graph by applying the above rules. If there is no closed circuit, there is no consistency problem since there is only one way to proceed from α1 to any given node. If there are closed circuits, one must make sure that one comes back to the same vector after one turn around any circuit. This can be arranged if the number of steps where one multiplies or divides by √ -- 2 (respectively, √ -- 3) is even.

Our construction shows that the lattice L is not unique. If there are only two different lengths for the lattice vectors μi, it is convenient to normalize the lengths so that the longest lattice vectors have length squared equal to two. This choice simplifies the factors B(μi,μj) 2B(μi,μi).

The rank 10 hyperbolic Coxeter groups are all crystallographic. The lattices preserved by E10 and DE 10 are unique up to an overall rescaling because the non-trivial m ij (i ⁄= j) are all equal to 3 and there is no choice in the ratios ci∕cj, all equal to one (first rule above). The Coxeter group BE10 preserves two (dual) lattices.

On the normalization of roots and weights in the crystallographic case

Since the vectors μi and αi are proportional, they generate identical reflections. Even though they do not necessarily have length squared equal to unity, the vectors μi are more convenient to work with because the lattice preserved by the Coxeter group is simply the lattice ∑ iℤ μi of points with integer coordinates in the basis {μi}. For this reason, we shall call from now on “simple roots” the vectors μi and, to follow common practice, will sometimes even rename them αi. Thus, in the crystallographic case, the (redefined) simple roots are appropriately normalized to the lattice structure. It turns out that it is with this normalization that simple roots of Coxeter groups correspond to simple roots of Kac–Moody algebras defined in the Section 4.6.3. A root is any point on the root lattice that is in the Coxeter orbit of some (redefined) simple root. It is these roots that coincide with the (real) roots of Kac–Moody algebras.

It is also useful to rescale the fundamental weights. The rescaled fundamental weights, of course proportional to ωi, are denoted Λi. The convenient normalization is such that

(μj|μj) (Λi|μj) = -------δij. (3.42 ) 2
With this normalization, they coincide with the fundamental weights of Kac–Moody algebras, to be considered in Section 4.


  Go to previous page Go up Go to next page