A Characterization of Local Nilpotence for Derivations of Ore Extensions

Abstract: Let $\Bbbk$ be an algebraically closed field of characteristic zero. A recent theorem of I. Pan characterizes the nonzero locally nilpotent derivations of $\Bbbk[x,y]$ in terms of their isotropy groups: such a derivation is locally nilpotent if and only if its isotropy group contains automorphisms of arbitrarily large degree. We study analogues of this characterization for several noncommutative algebras related to the affine plane. Our main result shows that Pan's characterization extends to the differential Ore extensions $A_h=\Bbbk[x][t;h(x)\partial_x]$, where $h\in\Bbbk[x]\setminus\Bbbk$. For the first Weyl algebra $A_1$, we establish the same equivalence for every nonzero locally finite derivation. For the quantum plane and the first quantum Weyl algebra, known results imply that the corresponding equivalence holds vacuously for nonzero derivations. In contrast, the converse fails for the free associative algebra $\Bbbk\langle x,y\rangle$: we construct a non-locally-nilpotent inner derivation whose isotropy group has unbounded degree. We further analyze this failure through the behavior of automorphisms and derivations under abelianization.