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- W1966654359 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 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 G L ( V , F ) . As a consequence of the answers to the above question, we offer a characterization of the real elements in G L ( 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 I ( 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 non-trivial for the unipotent elements in I ( V , B ) . 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 I ( V , B ) ." @default.
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- W1966654359 date "2011-01-01" @default.
- W1966654359 modified "2023-09-28" @default.
- W1966654359 title "On the existence of an invariant non-degenerate bilinear form under a linear map" @default.
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- W1966654359 doi "https://doi.org/10.1016/j.laa.2010.08.009" @default.
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