Preprint classifies locally finite derivations across three algebra families

The paper tackles a question from noncommutative algebra in a form that is easier to state than to solve: which derivations remain locally finite under repeated action? A derivation is an algebraic rule for how products change, while local finiteness asks whether its repeated action can be contained in finite-dimensional pieces. The answer is exact for three families of Ore extensions arising from the polynomial algebra k[x]: the quantum plane, the first quantum Weyl algebra and differential extensions.

The setup is abstract, but the question is precise

The authors identify the relevant Ore extensions up to isomorphism and examine the quantum plane, first quantum Weyl algebra and differential Ore extension separately through their defining relations. In the differential extension, the added generator t and polynomial generator x satisfy $tx - xt = h(x)$, where h(x) is the polynomial that defines the derivation. The proof strategy is algebraic, using theorem-and-proof arguments with leading terms, eigenvalues, degree growth and finite-dimensional invariant subspaces.

The quantum cases narrow quickly

In the quantum-plane case, the classification leaves only two basic directions. A derivation is locally finite exactly when it has the form $D = \lambda Dx + \mu Dy$. Here D is the derivation under study, Dx sends x to x and y to zero, Dy sends x to zero and y to y, and λ and μ are scalars from k, the base field. Every locally finite case is therefore a scalar combination of those two named derivations.

The paper also isolates inner derivations, which are associated with an element a of the quantum-plane algebra. The notation ada denotes the inner derivation associated with a, and the result is $ada\text{ is locally finite}\Longleftrightarrow a\in Z(A)$. In plain language, that derivation is locally finite exactly when a lies in Z(A), the center of the algebra A.

The first quantum Weyl algebra is more restrictive. Under the paper's stated condition on its parameter q, a locally finite derivation is exactly a scalar multiple of the Eulerian derivation H, written $D = cH$. Here D is the derivation, c is a scalar in k, and H is the Eulerian derivation defined in the paper. The locally nilpotent class, whose repeated action eventually becomes zero in the local sense, contains only the zero derivation. The notation $\mathrm{LND}(A1q(k)) = \{0\}$ records that conclusion, with A1q(k) denoting the algebra and LND denoting its locally nilpotent derivations.

That narrow result sits inside a fuller description. When q is not a root of unity, meaning no positive power of q equals 1, every derivation splits uniquely as $\mathrm{Der}(A1q(k)) = \mathrm{Inn}(A1q(k)) \oplus kH$. Here Der means all derivations, Inn means inner derivations, kH is the space of scalar multiples of H, and the direct-sum symbol records the unique split. Equivalently, each derivation has the form $D = adw + cH$, with adw the inner derivation associated with an algebra element w, determined modulo the center, and c a scalar. This full decomposition is stated only under the not-a-root-of-unity condition, whereas the locally finite classification is stated under the paper's q condition.

The defining polynomial changes the answer

The differential results turn on the polynomial h. The main classification assumes h is nonconstant; constant-h cases are treated separately through results described as already known. For the nonconstant cases, square-free means that the polynomial has no repeated factors, and the answer then splits again at degree 1 versus degree at least 2.

For square-free h of degree at least 2, local finiteness and local nilpotence coincide. The locally finite derivations are exactly $D = Dg(x)$, where D is the derivation and g(x) is a polynomial in k[x]. At degree 1, one extra component appears: $D = \lambda adt + Dg(x)$. Here λ is a scalar and adt is the inner derivation associated with t. The result identifies degree 1 as the place in this branch where that additional inner component enters.

The non-square-free branch follows a similar pattern, with a different named family. In its degree-at-least-2 case, the locally finite derivations are exactly $D = Dp(x)$, where p(x) is a polynomial in k[x], and local finiteness again coincides with local nilpotence. In the degree-1 case, the form is $D = \lambda Et + Dp(x)$. Et is the additional derivation arising in the non-square-free decomposition, and λ is a scalar.

A controlled collection, not a finite one

Those formulas classify individual derivations. The collection of locally finite derivations for the differential extension also has a structural description: for square-free nonconstant h, it is a solvable Lie subalgebra, a bracket-closed collection whose commutator structure simplifies in stages, and it is weakly locally finite. Yet it is not locally finite as a set of derivations. The non-square-free case has the same three properties, so the individual classifications do not turn the full collection into a locally finite set.

The boundary of the result

The supplied front matter identifies the work as arXiv:2608.28257v1, dated 28 Aug 2026, and labels it a preprint. Its conclusions are theorem-level classifications for algebraic objects over an algebraically closed field of characteristic zero, not empirical estimates from participants or observations. The result is therefore a precise map of the named algebra families within the stated mathematical setting, rather than a claim about behavior outside it.

Paper data and sources

Original title: On Locally Finite Derivations in Ore Extensions
Authors: R. Baltazar, A. Bianchi, M. Veloso, J. Schwarz
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text