Lattices and infinite-dimensional forms. `The lattice method'
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, \(S:=\{\partial \in k:{\bar \partial}=\partial \},\) \(T:=\{\partial +{\bar \partial}:\partial \in k\}.\) S/T carries a k-vector space structure by \(\lambda (\partial +T):=\lambda \partial {\bar \lambda}+T;\) let \(\| E\|\) denote the subspace of S/T generated by \(\{<x,x>+T:x\in E\}.\) Assume that \(\dim \| E\| \leq 1\), the trace-valued part of E is closed with respect to \(\bot\) and that all infinite-dimensional subspaces contain non-zero isotropic vectors. Then to any pair (E,F) where F is a linear subspace of E there is attached a list of fifteen objects (lattices, cardinal numbers,...) which are isometry invariants of the pair; moreover, this list gives a complete set of invariants (Theorem 5). The special case of k being a quaternion algebra over \({\mathbb{Z}}_ 2\), or a perfect field of characteristic 2 is considered. On the way to Theorem 5 the author examines indecomposable pairs (E,F), lists them up to isometry (p. 243-245ö), and proves several results concerning existence and uniqueness of their decomposition.
scientific article; zbMATH DE number 4047787
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