Extensions and contractions of the Lie algebra of q-pseudodifferential symbols on the circle

Journal of Functional Analysis

Volumn 143, Issue 1, 1997, Pages 55-97

  Khesin, Boris ^a   Lyubashenko, Volodymyr ^b   Roger, Claude ^c  

^a YALE UNIVERSITY   (United States)

^b UNIVERSITY OF YORK   (United Kingdom)

^c UNIVERSITÉ LYON 1   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030697545     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.1996.2896     Document Type: Article

References (41)

1. ∞ type, Comm. Math. Phys. 151 (1993), 233-243.
2. 2. L. A. Dickey, "Soliton Equations and Hamiltonian Systems," Advanced Series in Math. and Physics, Vol. 12, World Scientific, River Edge, NJ, 1991.
3. 3. A. S. Dhumaduldaev, Derivations and central extensions of the Lie algebras of pseudodifferential symbols, Algebra i Analiz 6, No. 1 (1994), 140-158.
4. 4. V. G. Drinfeld, Quantum groups, in "Proceedings of the ICM," Vol. 1, pp. 798-820, AMS, Providence, RI, 1987.
5. 5. D. B. Fairlie, P. Fletcher, and C. K. Zachos, Trigonometric structure constants for new infinite-dimensional algebras, Phys. Lett. B 218 (1989), 203-206.
6. 6. D. B. Fairlie, P. Fletcher, and C. K. Zachos, Infinite-dimensional algebras and a trigonometric basis for the classical Lie algebras, J. Math. Phys. 31, No. 5 (1990), 1088-1094.
7. 7. D. B. Fairlie and C. K. Zachos, Infinite-dimensional algebras, sine brackets, and SU(∞), Phys. Lett. B 224, Nos. 1-2 (1989), 101-107.
8. 8. B. L. Feigin, The Lie algebras gl(λ) and cohomologies of Lie algebras of differential operators, Russian Math. Surveys 43, No. 2 (1988), 169-170.
9. 9. D. B. Fuchs, "Cohomology of Infinite-Dimensional Lie Algebras," Nauka, Moscow, 1984.
10. 10. I. M. Gelfand and L. A. Dickii, Fractional powers of operators, and Hamiltonian systems, Funct. Anal. Appl. 10, No. 4 (1976), 13-29.
11. 11. I. M. Gelfand and D. B. Fuks, The cohomologies of the Lie algebra of the vector fields in a circle, Funct. Anal. Appl. 2, No. 4 (1968), 342-343.
12. 12. I. M. Gelfand and O. Mathieu, On the cohomology of the Lie algebra of Hamiltonian vector fields, J. Funct. Anal. 108 (1992), 347-360.
13. 13. M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, in "Deformation Theory of Algebras and Structures and Applications," Nato ASI Series, Vol. 247, Sect. 7.
14. 14. E. Getzler, Cyclic homology and Beilinson-Manin-Schechtman central extension, Proc. Amer. Math. Soc. 104, No. 3 (1988), 729-734.
15. ∞ and their vertex operator representations, Funct. Anal. Appl. 27, No. 1 (1993), 10-20.
16. 16. M. I. Golenishcheva-Kutuzova, private communication.
17. ∞ functions on a connected manifold, Lett. Math. Phys. 4, No. 3 (1980), 157-167.
18. 18. J. Hoppe, M. Olshanetsky, and S. Theisen, Dynamical systems on quantum tori Lie algebras, Comm. Math. Phys. 155, No. 3 (1993), 429-448.
19. 19. D. V. Juriev, Infinite dimensional geometry and quantum field theory of strings, III, preprint, ENS, Paris.
20. 20. V. G. Kac and D. M. Peterson, Spin and wedge representations of infinite dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA 78 (1981), 3308-3312.
21. 21. V. G. Kac and A. O. Radul, Quasifinite highest weight modules over the Lie algebra of differential operators on the circle, Comm. Math. Phys. 157, No. 3 (1993), 429-457.
22. 22. C. Kassel, Cyclic homology of differential operators, the Virasoro algebra and a q-analogue, Comm. Math. Phys. 146 (1992), 343-356.
23. 23. B. A. Khesin and I. S. Zakharevich, Poisson-Lie group of pseudodifferential operators and fractional KP-KdV hierarchies, C. R. Acad. Sci. Paris, Sér. I 315 (1993), 621-626; Poisson-Lie group of pseudodifferential operators, Comm. Math. Phys. 171, No. 3 (1995), 475-530.
24. B. A. Khesin and I. S. Zakharevich, Poisson-Lie group of pseudodifferential operators and fractional KP-KdV hierarchies, C. R. Acad. Sci. Paris, Sér. I 315 (1993), 621-626; Poisson-Lie group of pseudodifferential operators, Comm. Math. Phys. 171, No. 3 (1995), 475-530.
25. 24. A. A. Kirillov, La géométrie des moments pour les groupes de difféomorphismes, in "Progress in Math." (A. Connes et al., Eds.), Vol. 92, pp. 73-82, Birkhäuser, Boston, 1990.
26. 25. M. Kontsevich and S. Vishik, Determinants of elliptic pseudo-differential operators, preprint MPI/94-30 (1994).
27. 26. O. S. Kravchenko and B. A. Khesin, A nontrivial central extension of the Lie algebra of pseudodifferential symbols on the circle, Funct. Anal. Appl. 25, No. 2 (1991), 83-85.
28. 27. P. Lecomte, Applications of the cohomology of graded Lie algebras to formal deformations of Lie algebras, Lett. Math. Phys. 13 (1987), 157-166.
29. 28. P. Lecomte and C. Roger, Central extensions of the Lie algebras of Hamiltonian vector fields, manuscript in preparation.
30. 29. P. Lecompte and C. Roger, Modules et cohomologie des bigèbres de Lie, C. R. Acad. Sci. Paris Sér. I 310 (1990), 405-410.
31. 30. P. Lecomte and M. De Wilde, Formal deformations of the Poisson-Lie algebra of a symplectic manifold and star products. Existence, equivalence, derivations, in "Deformations Theory of Algebras and Structures," NATO-ASI Series C, Vol. 297, Kluwer Academic, Dordrecht.
32. 31. W.-L. Li, 2-cocycles on the Lie algebra of differential operators, J. Algebra 122 (1989), 64-80.
33. 32. F. Martinez Moras, J. Mas, and E. Ramos, Diffeomorphisms from higher dimensional W-algebras, Modern Phys. Lett. A 8, No. 23 (1993), 2189-2197.
34. ∞ and the Racah-Wigner algebra, Nucl. Phys. B 339 (1990), 191-221.
35. 34. A. Pressley and G. Segal, "Loop Groups," Cambridge University Press, Cambridge, 1984.
36. 35. C. Roger, Algèbres de Lie graduées et quantification, in "Symplectic Geometry and Mathematical Physics," Progress in Math., Vol. 99, Birkhäuser-Verlag, Boston/Basel, 1991.
37. 36. C. Roger, Extensions centrales d'algèbres et de groupes de Lie de dimension infinie, algèbre de Virasoro et généralisations, preprint, Inst. de Math., Univ. de Liège, 1993.
38. 37. M. V. Saveliev and A. M. Vershik, New examples of continuum graded Lie algebras, Phys. Lett. A 143, No. 3 (1990), 121-128.
39. 38. J. Vey, Déformation du crochet de Poisson sur une variété symplectique, Comment. Math. Helv. 50, No. 4 (1975), 421-454.
40. 39. M. Wambst, Homologie de l'algèbre quantique des symboles pseudo-différentiels sur le cercle, Université Louis Pasteur, Strasbourg, preprint.
41. 40. M. Wodzicki, Cyclic homology of differential operators, Duke Math. J. 54 (1987), 641-647.