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

Abstract

Let $A_n=\mathbb{k}[x_1^{\pm1},\ldots,x_n^{\pm1}]$ 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 $ρ(x_i)=λ_i x^{m_i}$, where $λ_i\in\mathbb{k}^\ast$, the exponent vectors $m_i$ are the columns of a matrix $M_ρ\in\operatorname{GL}_n(\mathbb Z)$, and $μ^TM_ρ=μ^T$. We also give an explicit parametrization in dimension two and relate these automorphisms to symmetries of diagonal flows on algebraic tori.