Ribbon 2-Knots, 1+1=2, and Duflo's Theorem for Arbitrary Lie Algebras
Dror Bar-Natan, Zsuzsanna Dancso, Nancy Scherich
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.
This paper is available as a
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It is also on the arXiv: arxiv.org/abs/1811.08558.