Neighborhood in the sense of Kneser for quaternionic lattices
type: Item
instance of: scholarly article
title: Neighborhood in the sense of Kneser for quaternionic lattices
zbMATH Open document ID: 0843.11022
DOI: 10.1007/BF02566012
author: Christine Bachoc
published in: Commentarii Mathematici Helvetici
publication date: 1996-08-13T00:00:00Z
full work available at URL: https://eudml.org/doc/140373
review text: 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.
Mathematics Subject Classification ID: 11E41
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 11H31
zbMATH DE Number: 838575
zbMATH Keywords: neighbour of an integral lattice
zbMATH Keywords: hermitian spaces over quaternion skew fields
zbMATH Keywords: connectedness of the neighbour graph
zbMATH Keywords: unimodular lattices over the Hurwitz order
reviewed by: Heinz-Georg Quebbemann
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 838575