Lattices and infinite-dimensional forms. `The lattice method'
type: Item
instance of: scholarly article
title: Lattices and infinite-dimensional forms. `The lattice method'
zbMATH Open document ID: 0643.06005
DOI: 10.1007/BF00337887
author: Herbert Gross
published in: Order
publication date: 1987-01-01T00:00:00Z
review text: Given a set M, a k-vector space E of infinite dimension, and a subgroup \({\mathcal G}\) of its automorphisms, the ``lattice method'' described by the author serves as a tool for classifying the \({\mathcal G}\)-orbits in M. The method consists in defining for any \(m\in M\) a ``structural lattice'' V(m) (i.e. a lattice equipped with some additional algebraic structure) which is an invariant of the orbit \({\mathcal G}m\). If the structural lattices are choosen appropriately, they will determine the orbits under some natural arithmetical assumptions. Application of this method yields a wealth of results for example in the following domains: (i) \textit{M. Studer} [Involutionen in abzählbardimensionalen alternierenden Räumen bei Charakteristik 2 (Ph. D. Thesis, Univ. Zürich) (1978), 11, p.221]. Assume k algebraically closed, \(char(k)=2\), E symplectic, dim E\(=\aleph_ 0\), and let M be the set of involutions of E, \({\mathcal G}\) the orthogonal group acting on M by conjugation. The orbits of this action are characterized by a set of seven cardinal number invariants. (ii) \textit{M. Wild} [Dreieckverbände: lineare and quadratische Darstellungstheorie (Ph. D. Thesis, Univ. Zürich) (1987)]. Assume that E is alternate, and an orthogonal sum of finite-dimensional subspaces, \(E^{\bot}=(O)\), and dim \(E\leq \aleph_ 3\). Let M be the set of subspaces of E, \({\mathcal G}\) the orthogonal group acting on M by evaluation. Orbits are then characterized by a set of 54 cardinal number invariants, using Schuppli's lattice \(V_ 3\) of 957 elements as structural lattice, cf. \textit{H. Gross}, \textit{Z. Lomecky}, and \textit{R. Schuppli} [Algebra Univers. 20, 267-291 (1985; Zbl 0574.06007)]. (iii) The main part of the paper is devoted to k-vector spaces E of countably infinite dimension (k a division ring of arbitrary characteristic) equipped with a non-degenerate Hermitean form \(<, >\). Let - denote the associated involutorial anti-automorphism of k, \(S:=\{\partial \in k:{\b…
reviewed by: Horst Szambien
Mathematics Subject Classification ID: 06C05
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 06A15
zbMATH DE Number: 4047787
zbMATH Keywords: orthogonal geometry
zbMATH Keywords: computation of lattices
zbMATH Keywords: lattice method
zbMATH Keywords: orbits
zbMATH Keywords: involutions
zbMATH Keywords: cardinal number invariants
zbMATH Keywords: orthogonal group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4047787