The classification of subspaces in Hermitean vector spaces
type: Item
instance of: scholarly article
title: The classification of subspaces in Hermitean vector spaces
zbMATH Open document ID: 0612.15018
DOI: 10.1016/0021-8693(87)90213-4
author: Herbert Gross
author: Remo Moresi
author: Christian Herrmann
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: This is another contribution to a field of bilinear algebra which flourished especially in the 1970's. The object of study here are pairs (E,F), where E is a finite-dimensional, nondegenerate \(\epsilon\)- Hermitian space over a division ring k with an antiautomorphism \(a\mapsto \bar a\), and F a k-linear subspace of E. As is well-known, Witt's theorem solves the classification problem in the case when all \(\epsilon\)- symmetric elements of k are traces \(a+\epsilon \bar a\). To settle also the exceptional case, the authors stress the role of the subspace \(E^*\) of trace-valued vectors in E. They introduce notions of orthogonal and isotypic decomposition which refer to E together with its two subspaces \(E^*\) and F, and to the lattice generated by all three. The main result then is a set of fourteen (almost independent, more or less complicated) invariants which determine the pair (E,F) up to isometry. To demonstrate its usefulness, known results on extension of isometries and congruence of subspaces in Hermitian spaces are derived.
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 11E16
zbMATH DE Number: 3989573
zbMATH Keywords: Hermitian form
zbMATH Keywords: isotypic decomposition
zbMATH Keywords: invariants
zbMATH Keywords: Hermitian spaces
reviewed by: Heinz-Georg Quebbemann
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90213-4
OpenAlex ID: W1998641597
cites work: Endomorphism algebras of vector spaces with distinguished sets of subspaces
cites work: On four subspaces of a vector space
cites work: Quadratische Räume mit zwei Unterräumen
cites work: Q5830680
cites work: Q5795141
cites work: Appendix: Degenerate bilinear forms
cites work: Bilinear forms on \(k\)-vectorspaces of denumerable dimension in the case of \(\text{char}(k) = 2\)
cites work: Q3871032
cites work: Quadratic forms in infinite dimensional vector spaces
cites work: FREE MODULAR LATTICES AND THEIR REPRESENTATIONS
cites work: Q3313912
cites work: Rahmen und erzeugende Quadrupel in modularen Verbänden
cites work: Classification of Countably Infinite Hermitian Forms Over Skewfields
cites work: Q5682390
cites work: On the Invariants of a Vector Subspace of a Vector Space over a Field of Characteristic Two
cites work: Zerlegung von S-Verbänden
cites work: Quadratic and Hermitian forms in additive and Abelian categories
cites work: On odd Hermitian forms
cites work: On the conjugacy classes in the unitary, symplectic and orthogonal groups
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description: scientific article; zbMATH DE number 3989573