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Two applications of elementary knot theory to Lie algebras and Vassiliev invariants. (English) Zbl 1032.57008

Let \(S(\mathfrak{g})\) and \(U(\mathfrak{g})\) be the symmetric algebra and the universal enveloping algebra of a Lie algebra \(\mathfrak{g}\) respectively. The Poincaré-Birkhoff-Witt map \(\chi: S(\mathfrak{g})\to U(\mathfrak{g})\) defines an isomorphism as vector spaces but not as algebras. The Duflo theorem states that the composition \(\chi\circ\partial_{j^{\frac{1}{2}}}\) is an algebra isomorphism for an isomorphism \(\partial_{j^{\frac{1}{2}}}: S(\mathfrak{g})\to S(\mathfrak{g})\) [see M. Duflo, Ann. Sci. Ecole norm. Sup., IV. Ser. 10, 265-288 (1977; Zbl 0353.22009)] .
On the other hand, let \(\mathcal{A}\) be the vector space spanned by Jacobi diagrams (uni-trivalent graphs) with a linear order of the set of univalent vertices (legs) modulo some relations, and \(\mathcal{B}\) the same space without order of legs. Note that both \(\mathcal{A}\) and \(\mathcal{B}\) have algebra structures with multiplications given by concatenation and disjoint union, respectively. Then the map \(\chi:\mathcal{B}\to\mathcal{A}\) defined by ‘averaging’ over all the possible orders of the legs is an isomorphism of vector spaces.
One of the main theorems of the paper is that an analog of the Duflo theorem holds for \(\mathcal{A}\) and \(\mathcal{B}\). To state it, we first define \(\Omega :=\exp\left(\sum_{n=1}^{\infty}b_{2n}\omega_{2n}\right)\in\mathcal{B}\), where \(b_{2n}\) is a ‘modified’ Bernoulli number and \(\omega_{2n}\) is the wheel with \(2n\) legs, that is the Jacobi diagram of a \(2n\)-gon with \(2n\) legs. Let \(\partial_{\Omega}:\mathcal{B}\to\mathcal{B}\) be the differential operator that sends a Jacobi diagram \(D\) to the sum of all ways of gluing all the legs of \(\Omega\) to some (or all) legs of \(D\) (it sends \(D\) to \(0\) if \(D\) has less legs than \(\Omega\)). Now it is proved that the composition \(\chi\circ\partial_{\Omega}:\mathcal{B}\to\mathcal{A}\) is an algebra isomorphism (Wheeling Theorem). The authors also show that \(\Omega\) is the Kontsevich integral of the unknot (Wheels Theorem).
As a corollary to the Wheeling Theorem they give a new proof of the Duflo theorem for metrized Lie (super-)algebras.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
17B20 Simple, semisimple, reductive (super)algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations

Citations:

Zbl 0353.22009

Software:

KnotTheory
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References:

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