Local Lie algebra structure on the escape divisor of the Alpöge map

Abstract

We study the pullback of the constant vector fields under Alpöge's polynomial map F=(P,Q,R): A³→A³ with constant Jacobian determinant −2. Near the generic point of the escape divisor t=0 (where t=1/z₁), we introduce a regular basis N, D₁, D₂ of the lifted derivations. Their exact commutation relations are computed: N generates a one-dimensional Lie ideal, and the quotient is an abelian Lie algebra of rank two. On the open set where a natural transversality condition holds, we construct explicit tangent vector fields T₁, T₂ that commute and span the induced tangent distribution. The special loci ry=1 and 3ry−2=0 are shown to have different geometric significance. The results provide a local differential-algebraic decomposition that resembles a bulk-boundary structure. The Alpöge map was announced by Levent Alpöge and attributed to Claude Fable 5. The escape divisor was recently studied by ulam.ai (2026). This work is, to our knowledge, the first to analyze the induced differential structure on this divisor. Includes the verification script verify_commutators_escape.py (SymPy) that checks all commutation relations and the regularity of the vector fields.