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Algebraic Lattices

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Convexity and Its Applications

Abstract

In his recent article [5], P. Gruber has surveyed the development of the Geometry of Numbers since the publication of the books of Cassels [3] and Lekkerkerker [6]. As none of these sources deals specifically with lattices other than those of Minkowski-type (i.e. a ℤ-module with N generators in ℝN) it seems worthwhile to trace the main developments there for lattices which have more algebraic structure. Even though these are often endowed with arithmetic properties as for example when ℝ and ℤ are replaced by some field k and a ring O of integral elements in k, we shall use the term ‘algebraic lattice’. A special case is the Leech lattice, which arose as a Z-module of rank 24 in ℝ24 and can now be interpreted in this way as an O-module of rank 12 in k12, where k is the Eisenstein field Q(ρ), ρ = exp(2πi/3). Such features of an algebraic lattice have in recent years been harnessed to deal with problems in other areas, e.g. Finite groups, Sphere packings and Codes. As we intend to keep mainly to the ideas and spirit of the Geometry of Numbers, the section devoted to these special algebraic lattices is brief and supplied only with a selection of general references. With this view, it is however appropriate to include details of the case when k is a field endowed with a non-archimedean valuation and, in particular, to survey the work of Armitage [1], [2] on the Riemann-Roch theorem. Although a number of other generalizations appear in the literature (e.g. Dubois [4]), we shall confine our attention to one sufficiently general to include most of these and which permits the study of non-commutative lattices. As a Minkowski lattice may be interpreted as a discrete subgroup Г of the additive group G of ℝN with the property that the factor group G/Г has compact closure, it is natural to review the impact of the ideas of the Geometry of Numbers on discrete subgroups of topological groups and of Lie groups. But, overall, the recurrent themes are the fundamental theorems of Minkowski ([7], [5]) (§3.1) for convex bodies and the compactness theorem of Mahler ([5], Section 2).

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References

Section 1

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, M5S 1Al, Canada

    J. H. H. Chalk

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  1. J. H. H. Chalk
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Editors and Affiliations

  1. Institut für Analysis, Technische Mathematik und Versicherungsmathematik, Technische Universität Wien, Gusshausstrasse 25-27, A-1040, Wien, Austria

    Peter M. Gruber

  2. Lehrstuhl für Mathematik II, Universität Siegen, Hölderlinstrasse 3, D-5900, Siegen 21, Germany

    Jörg M. Wills

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© 1983 Springer Basel AG

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Chalk, J.H.H. (1983). Algebraic Lattices. In: Gruber, P.M., Wills, J.M. (eds) Convexity and Its Applications. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5858-8_4

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  • DOI: https://doi.org/10.1007/978-3-0348-5858-8_4

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5860-1

  • Online ISBN: 978-3-0348-5858-8

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