The higher-order hom-associative Weyl algebras
Per Bäck
TL;DR
The paper shows that higher-order Weyl algebras in characteristic zero, classically formally rigid as associative algebras, admit nontrivial formal deformations as hom-associative algebras with twisting map $\alpha=\mathrm{id}$. It constructs these deformations as hom-associative iterated differential polynomial rings, establishes key structural properties (no zero divisors, simplicity, center behavior, and conditional power associativity), and determines their derivations and nuclei, culminating in a full isomorphism classification. It further demonstrates that these algebras form multi-parameter formal deformations of the classical Weyl algebras and induce corresponding hom-Lie deformations via the commutator, linking to deep conjectures such as the Generalized Dixmier Conjecture and the Jacobian Conjecture. The work thus bridges nonassociative deformation theory with longstanding questions in noncommutative algebra, while extending the deformation framework to a broad family of higher-order, hom-structured algebras.
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 nonzero 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 Generalized Dixmier Conjecture, and hence also to the Jacobian Conjecture.