Neighborhood in the sense of Kneser for quaternionic lattices
The well-known concept, due to M. Kneser, of a neighbour of an integral lattice on a quadratic space is applied in this paper to hermitian spaces over quaternion skew fields (ramified at the infinite places of a totally real number field). The author first proves the connectedness of the neighbour graph for lattices that are modular with respect to a given two-sided ideal of some maximal order. Then she specializes to unimodular lattices over the Hurwitz order and classifies them up to rank \(m=7\). When considered in euclidean space of dimension \(4m\), lattices of this type are even and isometric to their 2-scaled dual. The paper concludes with an example of such a lattice for \(m=8\) (defined, but non-integral over the Hurwitz order) that has minimum norm 6 and, therefore, gives the same 32-dimensional record sphere packing density as two examples constructed ten years ago by the reviewer. Computation of the automorphism groups by B. Souvignier has shown that these three lattices are pairwise non-isometric.
hermitian spaces over quaternion skew fields
unimodular lattices over the Hurwitz order
scientific article; zbMATH DE number 838575