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

Dror Bar-Natan, Zsuzsanna Dancso, Nancy Scherich

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 derive 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 Duflo's theorem in full generality, though via a lengthier path.