Extensions and contractions of the Lie algebra of q-pseudodifferential symbols (original) (raw)

Abstract

We construct cocycles on the Lie algebra of pseudo- and q-pseudodifferential symbols of one variable and on their close relatives: the sine-algebra and the Poisson algebra on two-torus. A ``quantum'' Godbillon-Vey cocycle on (pseudo)-differential operators appears in this construction as a natural generalization of the Gelfand-Fuchs 3-cocycle on periodic vector fields. We describe a nontrivial embedding of the Virasoro algebra

Key takeaways

sparkles

AI

  1. The construction introduces cocycles on the Lie algebra of q-pseudodifferential symbols, generalizing the Gelfand-Fuchs cocycle.
  2. The paper details a nontrivial embedding of the Virasoro algebra into the algebra of q-pseudodifferential operators.
  3. Cohomology groups of the algebra of q-pseudodifferential symbols span linearly the whole group, indicating rich structure.
  4. The study reveals that the group of q-pseudodifferential operators is solvable rather than unipotent, affecting integrability.
  5. It presents q-analogs of KP and KdV hierarchies, showing their complete integrability with infinite conserved quantities.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (38)

  1. I. Bakas, B. Khesin and E. Kiritsis, The logarithm the derivative operator and higher spin algebras of W ∞ type, Commun. Math. Phys. 151 (1993), 233-243.
  2. L. A. Dickey, "Soliton equations and Hamiltonian systems," Adv. Ser. Math. Phys. 12, World. Sci., River Edge, NJ, 1991.
  3. A. S. Dhumaduldaev, Derivations and central extensions of the Lie algebras of pseudodifferential symbols, Algebra i Analiz 6 (1994) n. 1, 140-158.
  4. V. G. Drinfeld, Quantum groups, in "Proceedings of the ICM," AMS, Providence, R.I. Vol. 1 (1987), 798-820.
  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. D. B. Fairlie, P. Fletcher and C. K. Zachos, Infinite-dimensional Algebras and a Trigono- metric Basis for the Classical Lie Algebras, J. Math. Phys. 31 (1990) n. 5, 1088-1094.
  7. D. B. Fairlie and C. K. Zachos, Infinite-dimensional Algebras, Sine Brackets, and SU (∞), Phys. Lett. B 224 (1989) n. 1-2, 101-107.
  8. B. L. Feigin, The Lie algebras gl(λ) and cohomologies of Lie algebras of differential operators, Russian Math. Surveys 43 (1988) n. 2, 169-170.
  9. D. B. Fuchs, "Cohomology of infinite-dimensional Lie algebras," Nauka, 1984.
  10. I. M. Gelfand and L. A. Dickii, Fractional powers of operators, and Hamiltonian systems, Funct. Anal. Appl. 10 (1976), n. 4, 13-29.
  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 (1968) n. 4, 342-343.
  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. M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, in "De- formation Theory of Algebras and Structures and Applications," Nato ASI Series 247, §7.
  14. E. Getzler, Cyclic homology and Beilinson-Manin-Schechtman central extension, Proc. Amer. Math. Soc. 104 (1988) n. 3, 729-734.
  15. M. I. Golenishcheva-Kutuzova and D. R. Lebedev, Z-graded trigonometric Lie subalgebras in Â∞ , B∞ , Ĉ∞ and D∞ and their vertex operator representations, Funct. Anal. Appl. 27 (1993) n. 1, 10-20.
  16. M. I. Golenishcheva-Kutuzova, private communication.
  17. S. Gutt, M. Cahen and M. de Wilde, Local cohomology of the algebra of C ∞ functions on a connected manifold, Lett. Math. Phys. 4 (1980) n. 3, 157-167.
  18. J. Hoppe, M. Olshanetsky and S. Theisen, Dynamical systems on quantum tori Lie algebras, Commun. Math. Phys. 155 (1993) n. 3, 429-448.
  19. D. V. Juriev, Infinite dimensional geometry and quantum field theory of strings III, preprint ENS, Paris.
  20. V. G. Kac and D. M. Peterson, Spin and wedge representations of infinite dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. USA, 78 (1981), 3308-3312.
  21. V. G. Kac and A. O. Radul, Quasifinite highest weight modules over the Lie algebra of differ- ential operators on the circle, Commun. Math. Phys. 157 (1993) n. 3, 429-457.
  22. C. Kassel, Cyclic homology of differential operators, the Virasoro algebra and a q-analogue, Commun. Math. Phys. 146 (1992), 343-356.
  23. B. A. Khesin and I. S. Zakharevich, Poisson-Lie group of pseudodifferential operators and fractional KP-KdV hierarchies, C. R. Acad. Sci. 315 Sér. I (1993), 621-626. Poisson-Lie group of pseudodifferential operators, preprint IHES/M/93/53, arXiv:hep-th/9312088, pp.66.
  24. A. A. Kirillov, La géométrie des moments pour les groupes de difféomorphismes, in Progress in Math., ed. A.Connes et al 92 (1990), 73-82.
  25. M. Kontsevich and S. Vishik, Determinants of elliptic pseudo-differential operators. Preprint.
  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 (1991) n. 2, 83-85.
  27. P. Lecomte, Applications of the cohomology of graded Lie algebras to formal deformations of Lie algebras, Lett. Math. Phys. 13 (1987), 157-166.
  28. P. Lecomte and C. Roger, Central extensions of the Lie algebras of Hamiltonian vector fields (in preparation).
  29. P. Lecomte and C. Roger, Modules et cohomologie des bigèbres de Lie, C. R. Acad. Sci. Paris 310 Série I (1990), 405-410.
  30. P. Lecomte and M. De Wilde, Formal deformations of the Poisson-Lie algebra of a symplec- tic manifold and star products. Existence, equivalence, derivations, in "Deformations theory of algebras and structures," NATO-ASI Series C 297, Kluwer Academic Publishers.
  31. W.-L. Li, 2-cocycles on the Lie algebra of differential operators, J. Algebra, 122 (1989), 64-80.
  32. F. Martinez Moras, J. Mas and E. Ramos, Diffeomorphisms from higher dimensional W - algebras, Modern Phys. Lett. A 8 (1993) n. 23, 2189-2197.
  33. C. N. Pope, L. J. Romans and X. Shen, W ∞ and the Racah-Wigner algebra, Nuclear Phys. B339 (1990), 191-221.
  34. A. Pressley and G. Segal, "Loop groups," Cambridge University Press, 1984.
  35. C. Roger, Algèbres de Lie graduées et quantification, in "Symplectic geometry and Mathematical Physics," Progress in Math. 99, Birkhäuser-Verlag, 1991.
  36. C. Roger, Extensions centrales d'algèbres et de groupes de Lie de dimension infinie, algèbre de Virasoro et generalisations, Preprint of Inst. de Math., Univ. de Liege, 1993.
  37. M. V. Saveliev and A. M. Vershik, New examples of continuum graded Lie algebras, Phys. Lett. A143 (1990) n. 3, 121-128.
  38. J. Vey, Deformation du crochet de Poisson sur une varieté symplectique, Comment. Math. Helv. 50 (1975) n. 4, 421-454.