Vector Fields and Automorphism Groups of Danielewski Surfaces

Abstract

AbstractIn 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.