Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T13:43:07.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Rank 1 preservers on the unitary Lie ring

Part of: Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

W. J. Wong
W. J. Wong
Affiliation:
University of Notre DameNotre Dame, Indiana 46556, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined. As an application, maps preserving commuting pairs of transformations are determined.

MSC classification

Secondary: 15A27: Commutativity
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 3 , December 1990 , pp. 399 - 417
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Dieudonné, J., ‘On the structure of unitary groups’, Trans. Amer. Math. Soc. 72 (1952), 367–385.CrossRefGoogle Scholar
[2]Dieudonné, J., La géométrie des groupes classiques, (Springer-Verlag, Berlin, 1955).Google Scholar
[3]Hahn, A. J., ‘Cayley algebras and the isomorphisms of the orthogonal groups over arithmetic and local domains’, J. Algebra 45 (1977), 210–246.CrossRefGoogle Scholar
[4]Marcus, M. and Moyls, B. N., ‘Transformations on tensor product spaces’, Pacific J. Math. 9 (1959), 1215–1221.CrossRefGoogle Scholar
[5]McDonald, B., R-linear endomorphisms of (R)n preserving invariants, (Mem. Amer. Math. Soc., no 287, Providence, R.I., 1983).Google Scholar
[6]Wall, G. E., ‘The structure of a unitary factor group’, Inst. Hautes Etudes Sc Publ Math. 1 (1959), 7–23.CrossRefGoogle Scholar
[7]Waterhouse, W. C., ‘Automorphisms of det(Xij): The group scheme approach’, Adv. in Math. 65 (1987), 171–203.CrossRefGoogle Scholar
[8]Waterhouse, W. C., ‘Linear transformations on self-adjoint matrices: The preservation of rank-one-plus-scalar’, Linear Algebra Appl. 74 (1986), 73–85.CrossRefGoogle Scholar
[9]Watkins, W., ‘Linear maps that preserve commuting pairs of matrices’, Linear Algebra Appl. 14 (1976), 29–35.CrossRefGoogle Scholar
[10]Wong, W. J., ‘Maps on simple algebras preserving zero products II: Lie algebras of linear type’, Pacific J. Math. 92 (1981), 469–488.CrossRefGoogle Scholar
[11]Wong, W. J., ‘Rank 1 preserving maps on linear transformations over noncommutative local rings’, J. Algebra 113 (1988), 263–293.CrossRefGoogle Scholar
[12]Wong, W. J., ‘Maps on spaces of linear transformations over semisimple algebras’, J. Algebra 115 (1988), 386–400.CrossRefGoogle Scholar