On the Isotropy Groups of Locally Finite Derivations of Laurent Polynomial Algebras

Let $A_n=\mathbb{k}[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ be a Laurent polynomial algebra over a field $\mathbb{k}$ of characteristic zero. We determine explicitly the automorphisms commuting with each locally finite derivation $$D_{\mu }=\sum _{i=1}^n{\mu }_i{x}_i\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i},\mu =({\mu }_1,\dots ,{\mu }_n{)}^T\in {\mathbb{k}}^n\mathrm{.}$$ These are precisely the automorphisms $\rho (x_i)={\lambda }_i{x}^{m_i}$, where ${\lambda }_i\in {\mathbb{k}}^{\ast }$, the exponent vectors $m_i$ are the columns of a matrix $M_{\rho }\in {\mathrm{GL}}_n(\mathbb{Z})$, and ${\mu }^T{M}_{\rho }={\mu }^T$. We also give an explicit parametrization in dimension two and relate these automorphisms to symmetries of diagonal flows on algebraic tori.