Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 088, 12 pages      arXiv:1604.08450      https://doi.org/10.3842/SIGMA.2016.088

A Duflo Star Product for Poisson Groups

Adrien Brochier
MPIM Bonn, Germany

Received May 18, 2016, in final form September 05, 2016; Published online September 08, 2016

Abstract
Let $G$ be a finite-dimensional Poisson algebraic, Lie or formal group. We show that the center of the quantization of $G$ provided by an Etingof-Kazhdan functor is isomorphic as an algebra to the Poisson center of the algebra of functions on $G$. This recovers and generalizes Duflo's theorem which gives an isomorphism between the center of the enveloping algebra of a finite-dimensional Lie algebra $\mathfrak{a}$ and the subalgebra of ad-invariant in the symmetric algebra of $\mathfrak{a}$. As our proof relies on Etingof-Kazhdan construction it ultimately depends on the existence of Drinfeld associators, but otherwise it is a fairly simple application of graphical calculus. This shed some lights on Alekseev-Torossian proof of the Kashiwara-Vergne conjecture, and on the relation observed by Bar-Natan-Le-Thurston between the Duflo isomorphism and the Kontsevich integral of the unknot.

Key words: quantum groups; knot theory; Duflo isomorphism.

pdf (362 kb)   tex (19 kb)

References

  1. Alekseev A., Enriquez B., Torossian C., Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations, Publ. Math. Inst. Hautes Études Sci. (2010), 143-189, arXiv:0903.4067.
  2. Alekseev A., Torossian C., The Kashiwara-Vergne conjecture and Drinfeld's associators, Ann. of Math. 175 (2012), 415-463, arXiv:0802.4300.
  3. Bar-Natan D., Le T.T.Q., Thurston D.P., Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. 7 (2003), 1-31, math.QA/0204311.
  4. Cartier P., Construction combinatoire des invariants de Vassiliev-Kontsevich des nøe uds, in R.C.P. 25, Vol. 45 (French) (Strasbourg, 1992-1993), Prépubl. Inst. Rech. Math. Av., Vol. 1993/42, Univ. Louis Pasteur, Strasbourg, 1993, 1-10.
  5. Cattaneo A.S., Felder G., Tomassini L., From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115 (2002), 329-352, math.QA/0012228.
  6. Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  7. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1989), 1419-1457.
  8. Drinfeld V.G., On quasitriangular quasi-Hopf algebras and on a group that is closely connected with ${\rm Gal}(\overline{\bf Q}/{\bf Q})$, Leningrad Math. J. 2 (1990), 829-860.
  9. Duflo M., Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci. École Norm. Sup. (4) 10 (1977), 265-288.
  10. Enriquez B., Halbout G., Quantization of quasi-Lie bialgebras, J. Amer. Math. Soc. 23 (2010), 611-653, arXiv:0804.0496.
  11. Etingof P., Kazhdan D., Quantization of Lie bialgebras. I, Selecta Math. (N.S.) 2 (1996), 1-41, q-alg/9506005.
  12. Kassel C., Turaev V., Chord diagram invariants of tangles and graphs, Duke Math. J. 92 (1998), 497-552.
  13. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, q-alg/9709040.
  14. Le T.T.Q., Murakami J., Representation of the category of tangles by Kontsevich's iterated integral, Comm. Math. Phys. 168 (1995), 535-562.
  15. Le T.T.Q., Murakami J., The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996), 41-64.
  16. Le T.T.Q., Murakami J., Parallel version of the universal Vassiliev-Kontsevich invariant, J. Pure Appl. Algebra 121 (1997), 271-291.
  17. Manchon D., Torossian C., Cohomologie tangente et cup-produit pour la quantification de Kontsevich, Ann. Math. Blaise Pascal 10 (2003), 75-106, math.QA/0106205.
  18. Semenov-Tian-Shansky M.A., Dressing transformations and Poisson group actions, Publ. Res. Inst. Math. Sci. 21 (1985), 1237-1260.
  19. Ševera P., Quantization of Lie bialgebras revisited, Selecta Math. (N.S.) 22 (2016), 1563-1581, arXiv:1401.6164.
  20. Tamarkin D.E., Another proof of M. Kontsevich formality theorem, math.QA/9803025.
  21. Tamarkin D.E., Operadic proof of M. Kontsevich's formality theorem, Ph.D. Thesis, The Pennsylvania State University, 1999.

Previous article  Next article   Contents of Volume 12 (2016)