Botta - Pierce - Watkins theorem

Let $\mathbb{F}$ be an arbitrary field, and let $n$ be a positive integer. Consider $\mathcal{M}_{n}(\mathbb{F}),$ the vector space of all $n\times n$ matrices over $\mathbb{F}.$ Define

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$\mathfrak{sl}_{n}(\mathbb{F})=\{A\in\mathcal{M}_{n}(\mathbb{F}):\,{\rm tr}(A)=% 0\},$
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$\mathcal{N}=\{A\in\mathcal{M}_{n}(\mathbb{F}):\,A\,\,\mbox{is nilpotent}\},$
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$\mathcal{GL}_{n}(\mathbb{F})=\{A\in\mathcal{M}_{n}(\mathbb{F}):\,\det(A)\neq 0\}.$

Notice that $\mathfrak{sl}_{n}(\mathbb{F})$ is a linear subspace of $\mathcal{M}_{n}(\mathbb{F})$ and $\mathcal{N}\subseteq\mathfrak{sl}_{n}(\mathbb{F}).$

The Botta – Pierce – Watkins theorem on linear preservers of the nilpotent matrices [BPW] can be formulated as follows.

Theorem 1

Let $\varphi:\mathfrak{sl}_{n}(\mathbb{F})\longrightarrow\mathfrak{sl}_{n}(\mathbb{% F})$ be a linear automorphism. Assume that $\varphi(\mathcal{N})\subseteq\mathcal{N}.$ Then either $\exists\,P\in\mathcal{GL}_{n}(\mathbb{F})\,\exists\,c\in\mathbb{F}\setminus\{0% \}\,\forall\,A\in\mathfrak{sl}_{n}(\mathbb{F}):\,\varphi(A)=cPAP^{-1},$ or $\exists\,P\in\mathcal{GL}_{n}(\mathbb{F})\,\exists\,c\in\mathbb{F}\setminus\{0% \}\,\forall\,A\in\mathfrak{sl}_{n}(\mathbb{F}):\,\varphi(A)=cPA^{\rm T}P^{-1}.$

The original proof is based on the Gerstenhaber - Serezhkin theorem, some elementary algebraic geometry, and the fundamental theorem of projective geometry.

References

BPW P. Botta, S. Pierce, W. Watkins, Linear transformations that preserve the nilpotent matrices, Pacific J. Math. 104 (No. 1): 39–46 (1983).

Title	Botta - Pierce - Watkins theorem
Canonical name	BottaPierceWatkinsTheorem
Date of creation	2013-03-22 19:20:21
Last modified on	2013-03-22 19:20:21
Owner	kammerer (26336)
Last modified by	kammerer (26336)
Numerical id	6
Author	kammerer (26336)
Entry type	Theorem
Classification	msc 15A04
Related topic	FundamentalTheoremOfProjectiveGeometry
Related topic	GerstenhaberSerezhkinTheorem