The explicit equivalence between the standard and the logarithmic star product for Lie algebras, I

Carlo A. Rossi

doi:10.1016/j.crma.2012.08.001

Lie Algebras/Mathematical Physics

The explicit equivalence between the standard and the logarithmic star product for Lie algebras, I

Presented by: Michèle Vergne

Carlo A. Rossi ¹

¹ MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany

Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 661-664.

Abstract
Résumé

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 $K \supset C$ 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 $K \supset C$ , 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.

Received: 2012-06-05
Accepted: 2012-08-06
Published online: 2012-07-01

DOI: 10.1016/j.crma.2012.08.001

Author's affiliations:

Carlo A. Rossi ¹

¹ MPIM Bonn, Vivatsgasse 7, 53111 Bonn, Germany

@article{CRMATH_2012__350_13-14_661_0,
     author = {Carlo A. Rossi},
     title = {The explicit equivalence between the standard and the logarithmic star product for {Lie} algebras, {I}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {661--664},
     publisher = {Elsevier},
     volume = {350},
     number = {13-14},
     year = {2012},
     doi = {10.1016/j.crma.2012.08.001},
     language = {en},
}

TY  - JOUR
AU  - Carlo A. Rossi
TI  - The explicit equivalence between the standard and the logarithmic star product for Lie algebras, I
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 661
EP  - 664
VL  - 350
IS  - 13-14
PB  - Elsevier
DO  - 10.1016/j.crma.2012.08.001
LA  - en
ID  - CRMATH_2012__350_13-14_661_0
ER  -

%0 Journal Article
%A Carlo A. Rossi
%T The explicit equivalence between the standard and the logarithmic star product for Lie algebras, I
%J Comptes Rendus. Mathématique
%D 2012
%P 661-664
%V 350
%N 13-14
%I Elsevier
%R 10.1016/j.crma.2012.08.001
%G en
%F CRMATH_2012__350_13-14_661_0

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/

References
Cited by

[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

Cited by Sources:

Comments - Policy