# Reflections

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ]. Lemma 4.8. Let M be a hyperbolic reﬂective lattice. For any isotropic vector v ∈ M the lattice v ⊥ /Zv is a deﬁnite reﬂective lattice. Another useful result is the following (see [18]). Theorem 4.9. Suppose a reﬂective hyperbolic lattice M decomposes as an orthogonal sum of a lattice M and a deﬁnite lattice K . Then M is reﬂective. Examples 4.10. 1) A lattice of rank 1 is always reﬂective. 2) The lattice U is reﬂective. The group O(U ) is ﬁnite. 3) All ﬁnite root lattices and their orthogonal sums are reﬂective. 4) An odd lattice In,1 = 1 n ⊥ −1 is reﬂective if and only if n ≤ 20. The Coxeter diagrams of their reﬂection groups can be found in [120], Chapter 6, §2 (n ≤ 18) and in [119] (n = 18, 19). Some of these diagrams are also discussed in 6 This deﬁnition is closely related but diﬀers from the deﬁnition of reﬂective hyperbolic lattices used in the works of V. Gritsenko and V. Nikulin on Lorentzian Kac-Moody algebras [52]. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 23 [23], Chapter 28. The lattices In,1 (2) are reﬂective for n ≤ 19 but 2-reﬂective only for n ≤ 9. For example, Figure 8 is the Coxeter diagram of the reﬂection group of the lattice I16,1 . • hhw • www • hh qqh wI qq q • •I  II  • I@   • •@ " @@ "" @ " @" •@ • • • @@ "" @I "" •I • " II •  Iw  • www • qq w qqq • hhhhh • • Figure 8. It is easy to see that the reﬂectivity property implies that all vertices not connected by the thick line correspond to roots α with (α, α) = 2 and the remaining two vertices correspond to roots with (α, α) = 1. Many other examples of Coxeter diagrams for 2-reﬂective lattices can be found in [84]. 5) Many examples (almost a classiﬁcation) of reﬂective lattices of ranks 3 and 4 can be found in [97], [103]. All even hyperbolic lattices of rank r > 2 for which Ref2 (M ) is of ﬁnite covolume (2-reﬂective lattices ) were found by V. Nikulin [84] (r = 4) and E. Vinberg (unpublished) (r = 4) (a survey of Nikulin’s results can be found in [29]). They exist only in dimension ≤ 19. Example 4.11. A hyperbolic lattice E2,3,r is 2-reﬂective if and only if 7 ≤ r ≤ 10. A hyperbolic lattice E2,4,r is 2-reﬂective if and only if r = 5, 6, 7. A hyperbolic lattice E3,3,r is 2-reﬂective if and only if r = 4, 5, 6. This easily follows from Proposition 2.4. The reﬂection groups of the lattices E2,3,7 , E3,3,4 and E2,4,5 are quasi-Lanner and coincide with the Coxeter groups W (2, 3, 7), W (3, 3, 4), W (2, 4, 5). The reﬂection groups of other lattices are larger than the corresponding groups W (p, q, r ). For example, the Coxeter diagram of Ref2 (E2,3,8 ) = Ref(E2,3,8 ) is the one in Figure 9. • • • • • • • • • • • •  • Figure 9. To prove this fact one ﬁrst observes, using Proposition 2.4, that the Coxeter diagram deﬁnes a group of coﬁnite vol...
View Full Document

## This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

Ask a homework question - tutors are online