# 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

• $\mathfrak{sl}_{n}(\mathbb{F})=\{A\in\mathcal{M}_{n}(\mathbb{F}):\,{\rm tr}(A)=% 0\},$

• $\mathcal{N}=\{A\in\mathcal{M}_{n}(\mathbb{F}):\,A\,\,\mbox{is nilpotent}\},$

• $\mathcal{GL}_{n}(\mathbb{F})=\{A\in\mathcal{M}_{n}(\mathbb{F}):\,\det(A)\neq 0\}.$

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 BottaPierceWatkinsTheorem 2013-03-22 19:20:21 2013-03-22 19:20:21 kammerer (26336) kammerer (26336) 6 kammerer (26336) Theorem msc 15A04 FundamentalTheoremOfProjectiveGeometry GerstenhaberSerezhkinTheorem