Lie Algebras/Mathematical Physics
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.

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.

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.

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DOI: 10.1016/j.crma.2012.08.001
Rossi, Carlo A. 1

1 MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany
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Rossi, Carlo A. 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. http://archive.numdam.org/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] Calaque, D.; Felder, G. Deformation quantization with generators and relations, J. Algebra, Volume 337 (2011), pp. 1-12 | DOI

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

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

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

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

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