In this paper, we study the Lie algebra of vector fields |${\operatorname{Vec}}(\textrm{D}_p)$| of a smooth Danielewski surface |$\textrm{D}_p$|⁠. We prove that the Lie subalgebra |$\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$| of |${\operatorname{Vec}}(\textrm{D}_p)$| generated by locally nilpotent vector fields is simple. Moreover, if the two Lie algebras |$\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$| and |$\langle{\operatorname{LNV}}(\textrm{D}_q) \rangle$| of two Danielewski surfaces |$\textrm{D}_p$| and |$\textrm{D}_q$| are isomorphic, then the surfaces |$\textrm{D}_p$| and |$\textrm{D}_q$| are isomorphic. As an application we prove that the ind-groups |${\operatorname{Aut}}(\textrm{D}_p)$| and |${\operatorname{Aut}}(\textrm{D}_q)$| are isomorphic if and only if |$\textrm{D}_p \simeq \textrm{D}_q$| as a variety. We also show that any automorphism of the ind-group |${\operatorname{Aut}}^\circ (\textrm{D}_p)$| is inner.

中文翻译：

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Danielewski曲面的向量场和自同构群

在本文中，我们研究向量域的李代数| $ {\ operatorname {Vec}}（\ textrm {D} _p）$ | 光滑的Danielewski表面| $ \ textrm {D} _p $ |⁠。我们证明了李子代数| $ \ langle {\ operatorname {LNV}}（\ textrm {D} _p）\ rangle $ | 的| $ {\ {operatorname VEC}}（\ {TEXTRM} d _p）$ | 由局部幂等矢量场生成的结果很简单。此外，如果两个李代数| $ \ langle {\ operatorname {LNV}}（\ textrm {D} _p）\ rangle $ | 和| $ \ langle {\ operatorname {LNV}}（（\ textrm {D} _q）\ rangle $ | 两个Danielewski曲面| $ \ textrm {D} _p $ | 和| $ \ textrm {D} _q $ | 是同构的，则表面| $ \ textrm {D} _p $ | 和| $ \ textrm {D} _q $ | 是同构的。作为应用程序，我们证明ind组| $ {\ operatorname {Aut}}（\ textrm {D} _p）$ | 和| $ {\ operatorname {Aut}}（\ textrm {D} _q）$ | 当且仅当| $ \ textrm {D} _p \ simeq \ textrm {D} _q $ |是同构的 各种各样。我们还显示ind组| $ {\ operatorname {Aut}} ^ \ circ（\ textrm {D} _p）$ |的任何同构。是内在的。