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Ribbon $2$–knots, $1+1=2$ and Duflo's theorem for arbitrary Lie algebras

Dror Bar-Natan, Zsuzsanna Dancso and Nancy Scherich

Algebraic & Geometric Topology 20 (2020) 3733–3760
Abstract

We explain a direct topological proof for the multiplicativity of the 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.

Keywords
knots, 2-knots, tangles, expansions, finite type invariants, Lie algebras, Duflo’s theorem
Mathematical Subject Classification 2010
Primary: 57M25
References
Publication
Received: 15 October 2019
Revised: 10 March 2020
Accepted: 26 March 2020
Published: 29 December 2020
Authors
Dror Bar-Natan
Department of Mathematics
University of Toronto
Toronto, ON
Canada
http://www.math.toronto.edu/~drorbn
Zsuzsanna Dancso
School of Mathematics and Statistics
The University of Sydney
Camperdown NSW
Australia
http://www.zsuzsannadancso.net
Nancy Scherich
Department of Mathematics and Statistics
Wake Forest University
Winston-Salem, NC
United States
http://www.nancyscherich.com