Let K be a field of characteristic zero, let k,k'∈ K^n each have exactly m nonzero elements, and let A_n^k and A_n^k' be the corresponding higher-order hom-associative Weyl algebras. Generalized Hom–Dixmier Conjecture. Every nonzero hom-associative K-algebra homomorphism A_n^k→ A_n^k' is a hom-associative K-algebra isomorphism. This is formulated as a family of conjectures indexed by m, motivated by the Generalized Dixmier Conjecture and the classification of these algebras up to isomorphism; its resolution is not given in the supplied text.

The Generalized Hom–Dixmier Conjecture for higher-order hom-associative Weyl algebras

From papers

Let $K$ be a field of characteristic zero, let $k,k'\in K^n$ each have exactly $m$ nonzero elements, and let $A_n^k$ and $A_n^{k'}$ be the corresponding higher-order hom-associative Weyl algebras. Generalized Hom–Dixmier Conjecture. Every nonzero hom-associative $K$ -algebra homomorphism

A_n^k\to A_n^{k'}

is a hom-associative $K$ -algebra isomorphism. This is formulated as a family of conjectures indexed by $m$ , motivated by the Generalized Dixmier Conjecture and the classification of these algebras up to isomorphism; its resolution is not given in the supplied text.

Progress summary

Partially solved

The full conjecture remains open, although a first-dimensional special case is known and its broader form has been linked to other major open problems.

The conjecture asserts that every nonzero homomorphism between matching higher-order hom-associative Weyl algebras must be an isomorphism. No proposer is identified in the supplied sources.

Known results

B. and J. Richter, 2020: the case $\mathrm{GHDC}_1^1$ is true.
Richter, 2019: every homomorphism between purely hom-associative first Weyl algebras is an isomorphism.
The higher-order paper shows that nonzero homomorphisms are automatically injective, so the conjecture reduces to surjectivity.
The stable family is equivalent to the Generalized Dixmier Conjecture and hence stably equivalent to the Jacobian Conjecture.

2024–2025 developments

A 2025 preprint formulates the full family but reports no proof or counterexample and states that the corresponding Generalized Dixmier Conjecture remains open for $n>1$ . It records Zheglov’s 2024 claim of a proof for the ordinary first Weyl algebra; that claim concerns only the first-dimensional case and is unverified.

Current status (as of August 2026): The first-dimensional special case is established, but the generalized conjecture for $n>1$ remains open; no verified proof or counterexample was found.

Sources

Sources & referencesView supporting material

Primary source

Per Bäck, “The higher-order hom-associative Weyl algebras”, arXiv:2502.04051 (2026).

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