Vector Fields and Automorphism Groups of Danielewski Surfaces

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.