Lattices in one-dimensional quaternionic linear space that are invariant with respect to finite linear groups

Abstract

i = j = k = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. We identify C with the subfield R+ Ri ofH. Let V be a right n-dimensional linear space over H. A finite-order element of GL(V ) is said to be a reflection if the fixed-point space of the element is (n− 1)-dimensional. All finite subgroups of GL(V ) generated by reflections are classified in [1]. Aiming at the extension to the quaternion case of the classification of the discrete groups generated by affine Hermitian reflections, which was obtained in [2], we arrive at the problem of describing, up to a homothety (i.e., up to multiplication by a nonzero element of H), the lattices in V that are invariant with respect to the finite subgroups of GL(V ) generated by reflections (by a lattice we mean a free subgroup of rank 4n in the additive group of V ). In this note, such a description is given for n = 1. This description is heavily used in the investigation of the general case, which will be treated elsewhere. Thus, in what follows, we assume that V is H with right multiplication by constants End(V ) is H acting on V by left multiplication, and GL(V ) is the multiplicative group H∗ of the skew field H. All nonidentity elements of finite order inH∗ are reflections. The description of finite subgroups ofH∗ is well known (see, e.g., [3]); it is given by the following proposition (below by 〈M〉 we denote the subgroup ofG generated by some subsetM ⊆ G). Proposition 1. Up to conjugacy, the finite subgroups of H∗ are exhausted by the following pairwise nonconjugate subgroups: