Subgroups of 𝐺𝐿(𝑛²,𝐂) containing 𝐏𝐒𝐔(𝐧)

Platonov, V.; Đoković, D.

doi:10.1090/S0002-9947-96-01466-3

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.

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