Subgroups of containing

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 be the image of the unitary group under the representation on the space of by complex matrices. We classify all connected Lie subgroups of containing . We use this result to obtain a description of all abstract overgroups of in .

We apply this classification to solve the problem of describing all invertible linear transformations of which preserve the set of normal matrices. Our results can be applied to solve many other problems of similar nature.

**1**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****2**Morikuni Goto and Frank D. Grosshans,*Semisimple Lie algebras*, Marcel Dekker, Inc., New York-Basel, 1978. Lecture Notes in Pure and Applied Mathematics, Vol. 38. MR**0573070****3**Morikuni Goto,*On an arcwise connected subgroup of a Lie group*, Proc. Amer. Math. Soc.**20**(1969), 157–162. MR**0233923**, 10.1090/S0002-9939-1969-0233923-X**4**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****5**V. P. Platonov and D. Z. Ðokovic,*Linear preserver problems and algebraic groups*, Preprint 94-029, Discrete Structures in Math., University of Bielefeld, 1994.**6***A survey of linear preserver problems*, Linear and Multilinear Algebra**33**(1992), 1--130.

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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:
https://doi.org/10.1090/S0002-9947-96-01466-3

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