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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 KacMoody algebras [52]. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 23 [23], Chapter 28. The lattices In,1 (2) are reﬂective for n ≤ 19 but 2reﬂective only
for n ≤ 9. For example, Figure 8 is the Coxeter diagram of the reﬂection group of
the lattice I16,1 .
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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 2reﬂective 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 (2reﬂective 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(unpublished) (r = 4) (a survey of Nikulin’s results can be found in [29]).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 2reﬂective if and only if 7 ≤ r ≤ 10.
A hyperbolic lattice E2,4,r is 2reﬂective if and only if r = 5, 6, 7. A hyperbolic
lattice E3,3,r is 2reﬂective 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
quasiLanner 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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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