Reflections

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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which Ref2 (M ) is of finite covolume (2-reflective lattices ) were found by V. Nikulin [84] (r = 4) and E. Vinberg (unpublished) (r = 4) (a survey of Nikulin’s results can be found in [29]).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...
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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.

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