Birational invariants of crystals and fields with a finite group of operators

Abstract

It is well known that an n-dimensional crystallographic group can be reconstructed from its point group, the integral representation of the point group which arises from its action on the translation lattice, and the 2-cocycle which glues the point group to the lattice ([2]). In practice, this constitutes a complicated list of invariants. When confronted with the classification of objects possessing a rich structure, the algebraic geometer first attempts to find more coarse birational invariants. We begin such a programme for torsion-free crystallographic groups. More precisely, if Γ is a torsion-free crystallographic group and k is a field then the group algebra k[Γ] is a non-commutative domain (see [6], chapter 13). It can be localized at its centre to yield a division algebra k(Γ) which is a crossed product; the Galois group is the point group and it acts on the rational function field generated by k and the lattice (regarded multiplicatively), which is a maximal subfield ([3]). What are thecommon invariants of Γ1 and Γ2 when k(Γ1) and k(Γ2) are isomorphic k-algebras?