向量场的李子代数与Jacobian猜想
Lie Subalgebras of vector fields and the Jacobian Conjecture
摘要：
研究了常散度仿射2-空间${\mathbb A}^{2}$的向量场$\mathrm{Vec}^{c}({\mathbb A}^{2})$的李子代数$L$，并对与${\mathbb A}^{2}$的仿射变换群$\mathrm{Aff}_{2}(K)$的李代数$\mathfrak{aff}_{2}$同构的$L$进行了分类.然后，我们证明了以下三个陈述是等价的：（i）雅可比猜想在2维中成立;（ii）所有同构于$\mathfrak{aff}_{2}$的李子代数$L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$在$\mathrm{Aut}({\mathbb A}^{2})$下是共轭的;（iii）所有同构于$\mathfrak{aff}_{2}$的李子代数$L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$是代数的。
英文摘要：
We study Lie subalgebras $L$ of the vector fields $\mathrm{Vec}^{c}({\mathbb A}^{2})$ of affine 2-space ${\mathbb A}^{2}$ of constant divergence, and we classify those $L$ which are isomorphic to the Lie algebra $\mathfrak{aff}_{2}$ of the group $\mathrm{Aff}_{2}(K)$ of affine transformations of ${\mathbb A}^{2}$. We then show that the following three statements are equivalent: (i) The Jacobian Conjecture holds in dimension 2; (ii) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are conjugate under $\mathrm{Aut}({\mathbb A}^{2})$; (iii) All Lie subalgebras $L \subset \mathrm{Vec}^{c}({\mathbb A}^{2})$ isomorphic to $\mathfrak{aff}_{2}$ are algebraic.
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