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

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

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).

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