The action of the symplectic group associated with a quadratic extension of fields
type: Item
instance of: scholarly article
title: The action of the symplectic group associated with a quadratic extension of fields
zbMATH Open document ID: 0935.12004
author: Claudio G. Bartolone
author: Maria Alessandra Vaccaro
published in: Journal of Algebra
publication date: 2000-05-11T00:00:00Z
review text: Let \(L\) be a field and \((V,f)\) a regular alternating space over \(L\). It is known that indecomposable subspaces, i.e., subspaces not splitting into the orthogonal sum of two proper subspaces, are either lines or hyperbolic planes. If \(K\) is a subfield of \(L\), then \(V\) is a vector space over \(K\). The authors ask the question, when is a \(K\)-subspace of \(V\) indecomposable? The classification of indecomposable \(K\)-subspaces allows the determination of the isometry classes in the set of \(K\)-subspaces of \(V\), which are orbits of the symplectic group \(\text{Sp}_L(V,f)\) in such a set. In this paper, the authors classify indecomposable \(K\)-spaces under the hypothesis that the extension \(L/K\) is quadratic. Since \(L/K\) is quadratic, every subspace \(W\) of \(V\) splits into the direct sum \(W= \text{comp}_L W\oplus W'\), where \(\text{comp}_LW\) denotes the largest \(L\)-subspace of contained in \(W\) and \(W'\) is a \(K\)-subspace of \(V\) (called a \(K\)-substructure) generated by vectors which are linearly independent over \(L\). Hence a pair \((m,n)\) of integers can be associated with \(W\), namely \(m= \dim_L \text{comp}_LW\) and \(n= \dim_KW'\). Thus \(\dim_KW= 2m+n\). \(K\)-subspaces of type \((m,0)\) and \((0,n)\) are \(L\)-subspaces and \(K\)-substructures, respectively. For a \(K\)-subspace \(W\) of type \((m,n)\) with \(m>0\), the authors prove that a necessary condition in order to which \(W\) is indecomposable is that \(m\leq 2\). Moreover, there is an indecomposable \(K\)-subspace of type \((1,n)\) just if \(n\not\equiv 0\pmod 4\) and an indecomposable \(K\)-subspace of type \((2,n)\) just if \(n\equiv 0\pmod 4\). Two indecomposable \(K\)-subspaces of the same type \((m,n)\), \(m=1,2\), are always isometric. Hence, in contrast with the case of \(K\)-substructures, the isometry classes in the set of indecomposable \(K\)-subspaces with nontrivial \(L\)-component do not depend on the field \(K\) and the number of isometry classes is p…
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 11E99
zbMATH DE Number: 1356294
zbMATH Keywords: quadratic field extension
zbMATH Keywords: regular alternating space
zbMATH Keywords: splitting
zbMATH Keywords: orthogonal sum
zbMATH Keywords: isometry classes
zbMATH Keywords: orbits
zbMATH Keywords: symplectic group
reviewed by: John N. Mordeson
MaRDI profile type: Publication
full work available at URL: jabr.1999.7921
OpenAlex ID: W2027454304
cites work: Q3999894
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cites work: Q3708857
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DOI: 10.1006/JABR.1999.7921
description: scientific article; zbMATH DE number 1356294