Kontsevich Deformation Quantization on Lie Algebras

Nabiha Ben Amar; Mouna Chaabouni; Mabrouka Hfaiedh

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 2, page 365-379
  • ISSN: 0392-4041

Abstract

top
We consider Kontsevich star product on the dual 𝔤 * of a general Lie algebra g equipped with the linear Poisson bracket. We show that this star product provides a deformation quantization by partial embeddings in the direction of the Poisson bracket.

How to cite

top

Ben Amar, Nabiha, Chaabouni, Mouna, and Hfaiedh, Mabrouka. "Kontsevich Deformation Quantization on Lie Algebras." Bollettino dell'Unione Matematica Italiana 10-B.2 (2007): 365-379. <http://eudml.org/doc/290385>.

@article{BenAmar2007,
abstract = {We consider Kontsevich star product on the dual $\frak\{g\}^*$ of a general Lie algebra g equipped with the linear Poisson bracket. We show that this star product provides a deformation quantization by partial embeddings in the direction of the Poisson bracket.},
author = {Ben Amar, Nabiha, Chaabouni, Mouna, Hfaiedh, Mabrouka},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {365-379},
publisher = {Unione Matematica Italiana},
title = {Kontsevich Deformation Quantization on Lie Algebras},
url = {http://eudml.org/doc/290385},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Ben Amar, Nabiha
AU - Chaabouni, Mouna
AU - Hfaiedh, Mabrouka
TI - Kontsevich Deformation Quantization on Lie Algebras
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/6//
PB - Unione Matematica Italiana
VL - 10-B
IS - 2
SP - 365
EP - 379
AB - We consider Kontsevich star product on the dual $\frak{g}^*$ of a general Lie algebra g equipped with the linear Poisson bracket. We show that this star product provides a deformation quantization by partial embeddings in the direction of the Poisson bracket.
LA - eng
UR - http://eudml.org/doc/290385
ER -

References

top
  1. ARNAL, D. - BEN AMAR, N., Kontsevich's Wheels and Invariant Polynomial Functions on the Dual of Lie Algebras, Lett. Math. Phys., 52 (2000), 291-300. Zbl1017.53073MR1796573DOI10.1023/A:1007610715981
  2. ARNAL, D. - BEN AMAR, N. - MASMOUDI, M., Cohomology of Good Graphs and Kontsevich Linear Star Products, Lett. Math. Phys., 48 (1999). Zbl0957.53046MR1709041DOI10.1023/A:1007618914771
  3. ARNAL, D. - CAHEN, M. - GUTT, S., Deformation on Coadjoint Orbits, J. Geom. Phys., 3 (1986), 327-351. Zbl0616.58013MR894630DOI10.1016/0393-0440(86)90013-6
  4. 1 BEN AMAR, N., Tangential Deformations on the Dual of Nilpotent Special Lie Algebras, Pacific Journal of Mathematics, 170, no. 2 (1995), 297-318. Zbl0847.17018MR1363867
  5. 2 BEN AMAR, N., K-Star Products on Dual of Lie Algebras, Journal of Lie Theory, 13 (2003), 329-357. Zbl1034.53096MR2003147
  6. 3 BEN AMAR, N., A Comparison between Rieffel's and Kontsevich's Deformation Quantizations for Linear Poisson Tensors, To appear in Pacific Journal of Mathematics. Zbl1155.53337MR2276500DOI10.2140/pjm.2007.229.1
  7. BAYEN, F. - FLATO, M. - FRONSDAL, C. - LICHNEROWICZ, A. - STERNHEIMER, D., Deformation Theory and Quantization, I, II, Ann. of Phys., 111 (1978), 61-151. Zbl0377.53025MR496157DOI10.1016/0003-4916(78)90224-5
  8. DE WILDE, M. - LECOMTE, P., Existence of Star Products and of Formal Deformations of the Poisson Lie Algebras of Arbitary Symplectic Manifold, Lett. Math. Phys., 7 (1983), 487-496. Zbl0526.58023MR728644DOI10.1007/BF00402248
  9. FEDESOV, B. F., A Simple Geometrical Construction of Deformation Quantization, J.D.G, 40 (1994), 213-238. MR1293654
  10. KONTSEVICH, M., Deformation Quantization of Poisson Manifolds, I. q-alg/9709040 (1997). 
  11. OMORI, H. - MAEDA, Y. - YOSHIOKA, A., Weyl Manifolds and Deformation Quantization, Adv. in Math., 85 (1991), 225-255. MR1093007DOI10.1016/0001-8708(91)90057-E
  12. PEDERSEN, G. K., C * -Algebras and their Automorphism Groups, London Math. Soc. Monographs, 14, Academic Press, London, (1979). MR548006
  13. 1 RIEFFEL, M. A.Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures, Amer. J. Math., 112 (1990), 657-686. Zbl0713.58054MR1064995DOI10.2307/2374874
  14. 2 RIEFFEL, M. A., Deformation Quantization of Heisenberg Manifolds, Comm. Math. Phys., 122 (1989), 531-562. Zbl0679.46055MR1002830
  15. 3 RIEFFEL, M. A., Continuous Fields of C * -Algebras Coming from Group Cocycles and Actions, Math. Ann., 283 (1989), 631-643. Zbl0646.46063MR990592DOI10.1007/BF01442857
  16. SHOIKHET, B., Vanishing of the Konsevich Integrals of the Wheels, Preprint q.alg/0007080. Zbl1018.53043MR1854132DOI10.1023/A:1010842705836
  17. VARADARAJAN, V.S., Lie Groups, Lie Algebras and their RepresentationsPrentice-Hall, New Jersey (1974). Zbl0371.22001MR376938
  18. WEINSTEIN, A., The Local Structure of Poisson Manifolds, J. Diff. Geom., 18 (1983) 523-557. Zbl0524.58011MR723816

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.