Ribbon 2-Knots, $1+1=2$, and Duflo's Theorem for Arbitrary Lie Algebras

Abstract

We explain a direct topological proof for the multiplicativity of Duflo isomorphism for arbitrary finite dimensional Lie algebras, and derive the explicit formula for the Duflo map. The proof follows a series of implications, starting with "the calculation 1+1=2 on a 4D abacus", using the study of homomorphic expansions (aka universal finite type invariants) for ribbon 2-knots, and the relationship between the corresponding associated graded space of arrow diagrams and universal enveloping algebras. This complements the results of the first author, Le and Thurston, where similar arguments using a "3D abacus" and the Kontsevich Integral were used to deduce Duflo's theorem for metrized Lie algebras; and results of the first two authors on finite type invariants of w-knotted objects, which also imply a relation of 2-knots with the Duflo theorem in full generality, though via a lengthier path.(∆ for doubling) of 2-knots in R 4 , which then, using an appropriate replacement of the Kontsevich integral, becomes an equality of arrow diagrams, which in itself can be interpreted as an equality in (a completion of) S(g * ) g ⊗ S(g * ) g ⊗ U (g) for an arbitrary finite-dimensional Lie algebra g, proving the multiplicative property of the Duflo isomorphism in full generality.