On the existence of an invariant non-degenerate bilinear form under a linear map
DSpace at IIT Bombay
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Title |
On the existence of an invariant non-degenerate bilinear form under a linear map
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Creator |
GONGOPADHYAY, K
KULKARNI, RS |
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Subject |
CONJUGACY CLASSES
REALITY PROPERTIES ALGEBRAIC-GROUPS Linear map Bilinear form Unipotents Real elements |
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Description |
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 T-invariant non-degenerate symmetric (resp. skew-symmetric) 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 non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group 1(V, B). A non-degenerate S-invariant subspace W of (V, B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant 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.
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Publisher |
ELSEVIER SCIENCE INC
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Date |
2012-06-26T07:35:08Z
2012-06-26T07:35:08Z 2011 |
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Type |
Article
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Identifier |
LINEAR ALGEBRA AND ITS APPLICATIONS,434(1)89-103
0024-3795 http://dx.doi.org/10.1016/j.laa.2010.08.009 http://dspace.library.iitb.ac.in/jspui/handle/100/14149 |
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Language |
English
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