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

Abstract

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 \[ D_μ=\sum_{i=1}^nμ_i x_i\frac{\partial}{\partial x_i}, \qquad μ=(μ_1,\ldots,μ_n)^T\in\Bbbk^n. \] 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.