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arxiv: 2502.04051 · v3 · pith:QW3BWIYDnew · submitted 2025-02-06 · 🧮 math.RA

The higher-order hom-associative Weyl algebras

Per B\"ack This is my paper

Pith reviewed 2026-05-23 04:31 UTC · model grok-4.3

classification 🧮 math.RA
keywords hom-associative algebrasWeyl algebrasformal deformationsDixmier conjectureJacobian conjecturedifferential polynomial ringssimplicity
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The pith

Higher-order Weyl algebras over characteristic zero fields admit nontrivial formal deformations as hom-associative algebras despite associative rigidity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that higher-order Weyl algebras, which resist nontrivial formal deformations when kept associative, gain new deformations once the associativity condition is relaxed to hom-associativity. These deformed algebras arise directly as hom-associative iterated differential polynomial rings. They turn out to be simple, free of zero divisors, and power-associative only in the cases that remain associative. The work computes their commuters, nuclei, centers, and derivations, classifies all such algebras up to isomorphism, and proves that a conjecture on their homomorphisms is stably equivalent to the Dixmier conjecture.

Core claim

Higher-order Weyl algebras over a field of characteristic zero can be formally deformed in a nontrivial way as hom-associative algebras. They arise as hom-associative iterated differential polynomial rings, contain no zero divisors, are simple, and are power-associative only when associative. Their commuters, nuclei, centers, and derivations are determined. All hom-associative Weyl algebras are classified up to isomorphism, and it is conjectured that nonzero homomorphisms between isomorphic ones are isomorphisms; this conjecture is stably equivalent to the Dixmier Conjecture and hence to the Jacobian Conjecture.

What carries the argument

The hom-associative deformation of the higher-order Weyl algebra, realized through hom-associative iterated differential polynomial rings.

If this is right

  • The algebras contain no zero divisors.
  • They are simple.
  • They are power-associative only when they remain associative.
  • All such algebras are classified up to isomorphism.
  • The homomorphism conjecture is stably equivalent to the Dixmier conjecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence to the Dixmier conjecture suggests that computational checks of low-order hom-associative Weyl algebras could indirectly test aspects of the Jacobian conjecture.
  • Similar hom-associative relaxations might produce nontrivial deformations for other known rigid associative algebras.
  • The classification up to isomorphism opens the possibility of enumerating all homomorphisms between concrete low-dimensional examples by direct matrix computation.

Load-bearing premise

The base field has characteristic zero.

What would settle it

An explicit zero divisor appearing in one of the constructed hom-associative Weyl algebras, or a nonzero non-surjective homomorphism between two isomorphic hom-associative Weyl algebras that is independent of the Dixmier conjecture, would falsify the corresponding claims.

read the original abstract

We show that the higher-order Weyl algebras over a field of characteristic zero, which are formally rigid as associative algebras, can be formally deformed in a nontrivial way as hom-associative algebras. We also show that these hom-associative Weyl algebras arise naturally as hom-associative iterated differential polynomial rings, that they contain no zero divisors, are power-associative only when associative, and that they are simple. We then determine their commuters, nuclei, centers, and derivations. Last, we classify all hom-associative Weyl algebras up to isomorphism and conjecture that all nonzero homomorphisms between any two isomorphic hom-associative Weyl algebras are isomorphisms. The latter conjecture turns out to be stably equivalent to the Dixmier Conjecture, and hence also to the Jacobian Conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that higher-order Weyl algebras over a field of characteristic zero are formally rigid as associative algebras but admit nontrivial formal deformations as hom-associative algebras. These hom-associative versions arise as hom-associative iterated differential polynomial rings, contain no zero-divisors, are simple, and are power-associative only when they are associative. The paper determines their commuters, nuclei, centers, and derivations, classifies all such algebras up to isomorphism, and conjectures that all nonzero homomorphisms between isomorphic ones are isomorphisms; this conjecture is shown to be stably equivalent to the Dixmier conjecture (and hence the Jacobian conjecture).

Significance. If the results hold, the work provides explicit constructions of nontrivial deformations in the hom-associative setting where associative rigidity holds, along with simplicity and classification results. The stable equivalence of the homomorphism conjecture to the Dixmier conjecture is a notable link between hom-associative algebra theory and classical open problems; the paper also supplies structural computations (centers, derivations) that may be reusable in related nonassociative settings.

minor comments (3)
  1. The introduction would benefit from a brief comparison table or explicit list distinguishing the higher-order hom-associative Weyl algebras from both ordinary Weyl algebras and standard hom-associative algebras, to clarify the new entities introduced.
  2. Notation for the twisting map and the higher-order parameter should be fixed consistently across sections; occasional shifts between subscript and superscript usage for the order index could confuse readers.
  3. The statement of the homomorphism conjecture (near the end) would be clearer if it explicitly restated the precise notion of 'stably equivalent' used in the equivalence proof to the Dixmier conjecture.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report, so we have no specific points requiring response or revision at this stage. We remain available to address any minor editorial suggestions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core contributions consist of explicit constructions of hom-associative deformations, iterated differential polynomial ring realizations, proofs of simplicity/no zero-divisors/power-associativity properties, computations of commuters/nuclei/centers/derivations, and an isomorphism classification. The final conjecture on homomorphisms is shown equivalent to the external Dixmier Conjecture rather than internally derived. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or self-citation chain; all load-bearing arguments rest on direct algebraic verification under the stated char-0 hypothesis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on standard definitions of Weyl algebras and hom-associativity from the literature; the main additions are the higher-order hom-associative instances and their properties. No free parameters or new entities with independent evidence are introduced in the abstract.

axioms (1)
  • domain assumption The base field has characteristic zero
    Stated explicitly as the setting in which the associative rigidity and the hom-associative deformations hold.
invented entities (1)
  • higher-order hom-associative Weyl algebra no independent evidence
    purpose: To realize nontrivial formal deformations of rigid associative algebras
    Constructed within the paper; the abstract provides no external falsifiable prediction or independent verification.

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Reference graph

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