Mathematicians Link Obscure Algebra Problem to Major Unsolved Conjecture
Researchers have discovered an unexpected connection between a new class of algebraic structures and the Jacobian Conjecture, one of mathematics' most famous open problems. The finding could reshape how mathematicians approach this decades-old puzzle, with potential implications for computational geometry and automated reasoning systems that rely on polynomial equations.
Originaltitel: The Higher-Order Hom-Associative Weyl 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 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.