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).
随便看	partially ordered ring PartialMapping partial mapping PartialOrder partial order PartialOrderingInATopologicalSpace partial ordering in a topological space PartialOrderWithChainConditionDoesNotCollapseCardinals partial order with chain condition does not collapse cardinals partialTiltingModule (partial) tilting module ParticleMovingOnACardioidAtConstantFrequency particle moving on a cardioid at constant frequency ParticleMovingOnTheAstroidAtConstantFrequency particle moving on the astroid at constant frequency Partition partition partition Partition1 PartitionedMatrix partitioned matrix PartitionFunction partition function PartitionIsEquivalentToAnEquivalenceRelation partition is equivalent to an equivalence relation