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Title:  On the existence of an invariant nondegenerate bilinear form under a linear map 
Authors:  GONGOPADHYAY, K KULKARNI, RS 
Keywords:  CONJUGACY CLASSES REALITY PROPERTIES ALGEBRAICGROUPS 
Issue Date:  2011 
Publisher:  ELSEVIER SCIENCE INC 
Citation:  LINEAR ALGEBRA AND ITS APPLICATIONS,434(1)89103 
Abstract:  Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F) > dim V. Let T V > V be an invertible linear map. We answer the following question in this paper. When does V admit a Tinvariant nondegenerate symmetric (resp. skewsymmetric) bilinear form ?We also answer the infinitesimal version of this question. Following Feit and Zuckerman 121, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V, F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V, F). Suppose V is equipped with a nondegenerate symmetric (resp. skewsymmetric) bilinear form B. Let S be an element in the isometry group 1(V, B). A nondegenerate Sinvariant subspace W of (V, B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper Sinvariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is nontrivial for the unipotent elements in I(V, B). The level of a unipotent T is the least integer k such that (T  1)(k) = 0. We also classify the levels of unipotents in I(V,B). (C) 2010 Elsevier Inc. All rights reserved. 
URI:  http://dx.doi.org/10.1016/j.laa.2010.08.009 http://dspace.library.iitb.ac.in/jspui/handle/100/14149 
ISSN:  00243795 
Appears in Collections:  Article

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