An explicit non-surjective endomorphism of the third Weyl algebra

Abstract

We construct an explicit endomorphism φ of the third Weyl algebra A₃(ℚ) that is injective but not surjective. The construction uses Alpöge's Keller map F=(P,Q,R) with constant Jacobian determinant −2. The pullbacks of the partial derivatives are defined via the transposed inverse Jacobian matrix, and we verify that they satisfy the Weyl relations. Injectivity follows from the simplicity of the Weyl algebra, and non-surjectivity is proved by a filtration argument together with the observation that F is not injective on ℂ³. The endomorphism is defined over ℚ and extends to A₃(ℂ). We also provide explicit formulas for (DF)⁻¹ and note that the divergence of the lifted derivations vanishes, making φ a *-endomorphism for the standard involution. The Alpöge map was announced by Levent Alpöge and attributed to Claude Fable 5. The idea of lifting this map to an endomorphism of the Weyl algebra was discussed on the Secret Blogging Seminar. This work provides the first fully explicit construction with closed-form expressions for (DF)⁻¹, symbolic verification of all Weyl relations, and a new proof of non-surjectivity using the order filtration. Includes the verification script verify_endomorphism_A3.py (SymPy) that checks: det(DF)=−2, B·DF=I₃, δᵢ(Fⱼ)=δᵢⱼ, [δᵢ,δⱼ]=0, collision points, symmetry F∘σ=τ∘F, and Piola identity.