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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Subgroups of $\operatorname{GL}(n^2,\mathbf{C})$ containing $\operatorname{PSU}(n)$


Authors: V. P. Platonov and D. Z. Ðokovic
Journal: Trans. Amer. Math. Soc. 348 (1996), 141-152
MSC (1991): Primary 20G20, 15A30
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Abstract: Let ${\operatorname{PSU}}(n)$ be the image of the unitary group ${\operatorname{U}}(n)$ under the representation $x\to axa^{-1}$ on the space $M_n({\mathbf C} )$ of $n$ by $n$ complex matrices. We classify all connected Lie subgroups of ${\operatorname{GL}}(n^2,{\mathbf C} )$ containing ${\operatorname{PSU}}(n)$. We use this result to obtain a description of all abstract overgroups of ${\operatorname{PSU}}(n)$ in ${\operatorname{GL}}(n^2,{\mathbf C} )$.

We apply this classification to solve the problem of describing all invertible linear transformations of $M_n({\mathbf C} )$ which preserve the set of normal matrices. Our results can be applied to solve many other problems of similar nature.


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Additional Information

V. P. Platonov
Affiliation: Department of Pure Mathematics, University of Waterloo Waterloo, Ontario, N2L 3G1 Canada
Email: dragomir@herod.uwaterloo.ca

D. Z. Ðokovic
Affiliation: Department of Pure Mathematics, University of Waterloo Waterloo, Ontario, N2L 3G1 Canada

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01466-3
PII: S 0002-9947(96)01466-3
Received by editor(s): August 6, 1994
Additional Notes: The first author was supported in part by NSERC Grant A-6197 and the Alexander von Humboldt Foundation
The second author was supported in part by NSERC Grant A-5285
Article copyright: © Copyright 1996 American Mathematical Society