Subgroups of $GL(n^2, \mathbf {C})$ containing $PSU(n)$
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- by V. P. Platonov and D. Ž. Đoković PDF
- Trans. Amer. Math. Soc. 348 (1996), 141-152 Request permission
Abstract:
Let $\mathrm {PSU}(n)$ be the image of the unitary group $\mathrm {U}(n)$ 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 $\mathrm {GL}(n^2,\mathbf {C} )$ containing $\mathrm {PSU}(n)$. We use this result to obtain a description of all abstract overgroups of $\mathrm {PSU}(n)$ in $\mathrm {GL}(n^2,\mathbf {C} )$. 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.References
- Dragomir Ž. Đoković and Vladimir P. Platonov, Algebraic groups and linear preserver problems, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 10, 925–930 (English, with English and French summaries). MR 1249362
- Morikuni Goto and Frank D. Grosshans, Semisimple Lie algebras, Lecture Notes in Pure and Applied Mathematics, Vol. 38, Marcel Dekker, Inc., New York-Basel, 1978. MR 0573070
- Morikuni Goto, On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc. 20 (1969), 157–162. MR 233923, DOI 10.1090/S0002-9939-1969-0233923-X
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- V. P. Platonov and D. Ž. Đoković, Linear preserver problems and algebraic groups, Preprint 94-029, Discrete Structures in Math., University of Bielefeld, 1994.
- A survey of linear preserver problems, Linear and Multilinear Algebra 33 (1992), 1–130.
Additional Information
- V. P. Platonov
- Affiliation: Department of Pure Mathematics, University of Waterloo Waterloo, Ontario, N2L 3G1 Canada
- Email: dragomir@herod.uwaterloo.ca
- D. Ž. Đoković
- Affiliation: Department of Pure Mathematics, University of Waterloo Waterloo, Ontario, N2L 3G1 Canada
- 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 - © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 141-152
- MSC (1991): Primary 20G20, 15A30
- DOI: https://doi.org/10.1090/S0002-9947-96-01466-3
- MathSciNet review: 1321586