Comptes Rendus
Lie Algebras/Mathematical Physics
The explicit equivalence between the standard and the logarithmic star product for Lie algebras, I
[Une équivalence explicite entre les produit-étoilés standard et logarithmique pour une algèbre de Lie, I]
Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 661-664.

Dans cette note, on construit explicitement une équivalence entre les deux produits-étoilés ⋆ et log sur lʼalgèbre symétrique S(g) associée à une algèbre de Lie g de dimension finie sur un corps KC, construits en utilisant le propagateur angulaire standard et le propagateur logarithmique respectivement : lʼoperateur differentiel dʼordre infini à coéfficients constants réalisant cette équivalence est relié à lʼincarnation du groupe de Grothendieck–Teichmüller considérée par Kontsevich (1999) dans [5, Theorem 7]. On présente dans cette première partie le résultat principal, dont la démonstration sera donnée dans la deuxième partie.

The purpose of this note is to establish an explicit equivalence between two star products ⋆ and log on the symmetric algebra S(g) of a finite-dimensional Lie algebra g over a field KC associated with the standard angular propagator and the logarithmic one respectively: the differential operator of infinite order with constant coefficients realizing the equivalence is related to the incarnation of the Grothendieck–Teichmüller group considered by Kontsevich (1999) in [5, Theorem 7]. We present in the first part the main result, and devote the second part to its proof.

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DOI : 10.1016/j.crma.2012.08.001

Carlo A. Rossi 1

1 MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany
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Carlo A. Rossi. The explicit equivalence between the standard and the logarithmic star product for Lie algebras, I. Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 661-664. doi : 10.1016/j.crma.2012.08.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.08.001/

[1] A. Alekseev, J. Löffler, C.A. Rossi, C. Torossian, Stokesʼ Theorem in presence of poles and logarithmic singularities, in preparation.

[2] A. Alekseev, J. Löffler, C.A. Rossi, C. Torossian, The logarithmic formality quasi-isomorphism, in preparation.

[3] D. Calaque; G. Felder Deformation quantization with generators and relations, J. Algebra, Volume 337 (2011), pp. 1-12 | DOI

[4] D. Calaque; G. Felder; A. Ferrario; C.A. Rossi Bimodules and branes in deformation quantization, Compos. Math., Volume 147 (2011) no. 1, pp. 105-160 | DOI

[5] M. Kontsevich Operads and motives in deformation quantization, Lett. Math. Phys., Volume 48 (1999) no. 1, pp. 35-72 | DOI

[6] M. Kontsevich Deformation quantization of Poisson manifolds, Lett. Math. Phys., Volume 66 (2003) no. 3, pp. 157-216

[7] C.A. Rossi The explicit equivalence between the standard and the logarithmic star product for Lie algebras, II, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012) | DOI

[8] B. Shoikhet Vanishing of the Kontsevich integrals of the wheels, Lett. Math. Phys., Volume 56 (2001) no. 2, pp. 141-149 | DOI

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