Classification of 2-reflective hyperbolic lattices of rank 4
Classification of 2-reflective hyperbolic lattices of rank 4
复制标题
4 阶 2 反射双曲格子的分类
DOI：
10.1090/s0077-1554-07-00160-4
复制
发表时间：
2007
影响因子：
--
通讯作者：
V. V. Nikulin
中科院分区：
文献类型：
--
作者：
Trudy Moskov;E. Vinberg;V. V. Nikulin
一个双曲格称为2-反射格，如果它的自同构群包含一个由2-反射生成的有限指数子群。我们确定了秩为4的所有20个反射双曲格。(For所有其他等级的值，这是由V. V. Nikulin完成的。在本文中，我们分类秩为4的2-反射双曲格（定义见下文）。作者于1981年获得这一分类。虽然在[7]中提到过，但这些证明仅在1998年作为预印本出版[14]。2-反射双曲格的分类是具有有限自同构群的代数K3-曲面分类的关键部分[5，6]。一个二次格是一个自由阿贝尔群，它具有一个非退化的积分对称双线性形式，称为标积。如果标积是正定的，则称它是欧几里得的;如果它有签名（n，1），则称它是双曲的。一个二次格L可以看作是一个（伪）欧几里得向量空间V = L <$R中的格。于是L的自同构群O（L）变成一个格（即，有限余体积的离散子群）在（伪）正交群O（V）中。在双曲情形下，双曲面（0.1）（x，x）= −1的两个连通分支之一可以看作n维罗巴切夫斯基空间L的模型，使得L的运动群是O（V）中指数为2的子群O′（V），它由保持双曲面（0.1）的每个连通分支的变换组成。在这个模型中，L的平面是双曲面L与V的子空间的非空交点。L中无穷远点对应于V的各向同性一维子空间。群O′（L）= O（L）<$O′（V）是L的有限余体积运动的离散群。如果（e，e）= k > 0且（0.2）2（e，x）∈ kZ <$x ∈ L，则称二次格L的本原向量e为根，或者更精确地说，称k-根。任何根e都产生正交反射Re：x → x − 2（e，x）（e，e）e，它保持L，即，属于O（L）。在双曲的情况下，Re在超平面He = {x ∈ L：（e，x）= 0} 2000数学主题分类中产生反射。第11 H 06 2007年美国数学学会American Mathematical Society
A hyperbolic lattice is said to be 2-reflective if its automorphism group contains a subgroup of finite index generated by 2-reflections. We determine all 2reflective hyperbolic lattices of rank 4. (For all other values of the rank, this was done by V. V. Nikulin.) In this paper we classify 2-reflective hyperbolic lattices of rank 4 (for definitions, see below). The author obtained this classification in 1981. While it was mentioned in [7], the proofs were only published as a preprint [14] in 1998. The classification of 2-reflective hyperbolic lattices is a pivotal part of the classification of algebraic K3-surfaces with finite automorphism groups [5, 6]. A quadratic lattice is a free abelian group endowed with a non-degenerate integral symmetric bilinear form, called the scalar product. It is said to be Euclidean if the scalar product is positive definite and hyperbolic if it has signature (n, 1). A quadratic lattice L can be viewed as a lattice in a (pseudo-)Euclidean vector space V = L ⊗ R. The automorphism group O(L) of L then becomes a lattice (i.e., a discrete subgroup of finite co-volume) in the (pseudo-)orthogonal group O(V ). In the hyperbolic case, one of the two connected components of the hyperboloid (0.1) (x, x) = −1 can be viewed as a model of n-dimensional Lobachevsky space L such that the group of motions of L is the subgroup O′(V ) of index 2 in O(V ) consisting of transformations preserving each connected component of the hyperboloid (0.1). In this model, the planes of L are non-empty intersections of the hyperboloid L with subspaces of V. The points at infinity in L correspond to the isotropic one-dimensional subspaces of V. The group O′(L) = O(L) ∩ O′(V ) is a discrete group of motions of L with finite co-volume. A primitive vector e of the quadratic lattice L is called a root or, more precisely, a k-root if (e, e) = k > 0 and (0.2) 2(e, x) ∈ kZ ∀x ∈ L. Any root e gives rise to an orthogonal reflection Re : x → x − 2(e, x) (e, e) e, which preserves L, i.e., belongs to O(L). In the hyperbolic case, Re gives rise to a reflection in the hyperplane He = {x ∈ L : (e, x) = 0} 2000 Mathematics Subject Classification. Primary 11H06. c ©2007 American Mathematical Society