On the existence of an invariant non-degenerate bilinear form under a linear map

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• 作者：Krishnendu Gongopadhyay ; Ravi S. Kulkarni
• 关键词：Linear map ; Bilinear form ; Unipotents ; Real elements
• 刊名：Linear Algebra and its Applications
• 出版年：2011
• 出版时间：1 January 2011
• 年：2011
• 卷：434
• 期：1
• 页码：89-103
• 全文大小：236 K

文摘

Let be a vector space over a field . Assume that the characteristic of is large, i.e. . Let be an invertible linear map. We answer the following question in this paper. When does 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 2, 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 . As a consequence of the answers to the above question, we offer a characterization of the real elements in .

Suppose is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group . A non-degenerate S-invariant subspace of 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 non-trivial for the unipotent elements in . The level of a unipotent T is the least integer k such that (T-I)^k=0. We also classify the levels of unipotents in .