On the morphism of Duflo-Kirillov type. (English) Zbl 1134.53304
From the text: We give the detailed proofs of some of M. Kontsevich’s claims in the paper “Deformation quantization of Poisson manifolds. I” [Lett. Math. Phys. 66, No. 3, 157–216 (2003; Zbl 1058.53065)], i.e., we prove the compatibility of the two cup products, and prove two conjectures by using the formalism of the proof of Kontsevich’s proof of his Formality theorem; the conjecture of Raïs, Kashiwara and Vergne and the conjecture of D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. P. Thurston [Isr. J. Math. 119, 217–237 (2000; Zbl 0964.57010)].
These conjectures are connected with the Duflo-Kirillov isomorphism relating the center of the universal enveloping algebra \(U(\mathfrak{g})\) of a finite-dimensional Lie algebra \(\mathfrak g\) with the algebra of invariants \((\text{Sym}(\mathfrak g))^{\mathfrak g}\). These objects can be realized by differential operators. Kontsevich’s proof of the formality theorem is imprecise about the correct sign. Moreover, we calculate how the sign appears.
These conjectures are connected with the Duflo-Kirillov isomorphism relating the center of the universal enveloping algebra \(U(\mathfrak{g})\) of a finite-dimensional Lie algebra \(\mathfrak g\) with the algebra of invariants \((\text{Sym}(\mathfrak g))^{\mathfrak g}\). These objects can be realized by differential operators. Kontsevich’s proof of the formality theorem is imprecise about the correct sign. Moreover, we calculate how the sign appears.
MSC:
| 53D55 | Deformation quantization, star products |
| 17B35 | Universal enveloping (super)algebras |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
References:
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