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CHARACTERISTIC CLASSES OF FLAGS OF FOLIATIONS AND LIE ALGEBRA COHOMOLOGY

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We prove the conjecture by Feigin, Fuchs, and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given ag at the origin. The latter encodes characteristic classes of ags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan.

Feigin, Fuchs, and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory. By the application of degeneration theorems of appropriate Hochschild-Serre spectral sequences, we avoid the need to use the methods of FFG, and moreover, we are able to describe all the symmetric powers at once.

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References

  1. И. Н. Бернштейн, Б. И. Розенфельд, Однородные пространства бесконeчномерных алгебр Ли и характеристические классы слоений, УМН 28 (1973), вып.. 4(172), 103-138. Engl. transl.: I. N. Bernstein, B. Rosenfeld, Homogeneous spaces of infinite-dimensional Lie algebras and characteristic classes of foliations, Russian Math. Surveys 28 (1973), no. 4, 107-142.

  2. И. Н. Бернштейн, И. М. Гельфанд, С. И. Гельфанд, Структура представлений, порождённых векторами стархего веса, Функц. анализ и его прил. 5 (1971), вып. 1, 1-9, Engl. transl.: I. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand, Structure of representations generated by vectors of highest weight, Funct. Analysis Appl. 5 (1971), no. 1, 1-8.

  3. R. Bott, Lectures on characteristic classes and foliations, Notes by Lawrence Conlon, with two appendices by J. Stasheff, in: Lectures on Algebraic and Differential Topology, (Second Latin American School in Math., Mexico City 1971), Lecture Notes in Mathematics, Vol. 279, Springer, Berlin, 1972, pp. 1-94.

  4. R. Bott, G. Segal, The cohomology of vector fields on a manifold, Topology 16 (1977), 285-298.

    Article  MathSciNet  MATH  Google Scholar 

  5. N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Translated from the 1968 French original by Andrew Pressley, Springer-Verlag, Berlin, 2002.

  6. A. Candel, L. Conlon, Foliations, II, Graduate Studies in Mathematics, Vol. 60, Amer. Math. Soc., Providence, RI, 2003.

  7. H. Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, in: Colloque de Topologie (Espaces Fibrés), Bruxelles, 1950, Georges Thone, Liége, Masson et Cie., Paris, 1951, pp. 57-71.

  8. В. В. Доценко, Гомологии алгебры Ли векторных полей на прямой с коэффициентами в симметрических степенях её присоединённого представления, Функц. анализ и eго пpил. 40 (2006), вып. 2, 13-19, Engl. transl.: V. Dotsenko Homology of the Lie algebra of vector fields on the line with coefficients in symmetric powers of its adjoint representation, Funct. Analysis Appl. 40 (2006), no. 2, 91-96.

  9. M. Engeli, G. Felder, A Riemann-Roch-Hirzebruch formula for traces of differential operators, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 4, 621-653.

  10. Б. Л. Фейгин, Характеристические классы флагов слоения, Функц. Анализ и eго пpил 9 (1975), вып. 4, 49-56, Engl. transl.: B. L. Feigin, Characteristic classes of flags of foliation, Funct. Analysis Appl. 9 (1975), no. 4, 312-317.

  11. B. L. Feigin, B. L. Tsygan, Riemann-Roch theorem and Lie algebra cohomology, in: Proc. of the Winter School on Geometry and Physics (Srni, 9-16 January 1988), Rend. Circ. Mat. Palermo (2) Suppl. 21 (1989), 15-52.

  12. W. Fulton, J. Harris, Representation Theory, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.

  13. Д. Б. Фукс, Когомологии Бесконечномерных алгебр Ли, Наука, M., 1984. Engl. transl.: D. B. Fuks, Cohomology of Infinite-dimensional Lie Algebras, Con-temporary Soviet Mathematics, Consultants Bureau, New York, 1986.

  14. B. Feigin, G. Felder, B. Shoikhet, Hochschild cohomology of the Weyl algebra and traces in deformation quantization, Duke Math. J. 127 (2005), 487-517.

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Feigin, A. Losev, B. Shoikhet, Riemann-Roch-Hirzebruch theorem and topological quantum mechanics, arXiv:math.QA/0401400.

  16. I. M. Gelfand, The cohomology of infinite dimensional Lie algebras: some questions of integral geometry, in: Actes du Congrès International des Mathématiciens (Nice, 1970), t. 1, Gauthier-Villars, Paris, 1971, 95-111.

  17. И. М. Гельфанд, Д. А. Каждан, Д. Б. Фукс, Действия бесконечномерных алгебр Ли, Функц. Анализ и eго пpил. 6 (1972), вып. 1, 10-15, Engl. transl.: I. M. Gel'fand, D. A. Kazhdan, D. B. Fuks, The actions of infinite-dimensional Lie algebras, Funct. Analysis Appl. 6 (1972), no. 1, 9-13.

  18. И. М. Гельфанд, Д. Б. Фукс, Когомологии алгебры Ли формальных векторных полей, Изв. АН СССР, Сер. мат. 34 (1970), вып. 2, 322-337. Engl. transl.: I. M. Gelfand; D. B. Fuks; Cohomology of the Lie algebra of formal vector fields, Math. USSR-Izv. 4 (1970), no. 2, 327-342.

  19. И. М. Гельфанд, Д. А. Каждан, Д. Б. Фукс, Когомологии алгебры Ли формальных векторных полей с коeффициентами в сопряжëнном с ней пространстве и вариации характеристических классов слоений, Функц. Анализ и eгo пpил. 8 (1974), вып. 2, 13-29, Engl. transl.: I. M. Gel'fand, B. L. Feigin, D. B. Fuks, Cohomologies of the Lie algebra of formal vector fields with coefficients in its adjoint space and variations of characteristic classes of foliations, Funct. Analysis Appl. 8 (1974), no. 2, 99-112.

  20. G. Hochschild, J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591-603.

  21. A. Gorodentsev, A. Khoroshkin, A. Rudakov, On syzygies of highest weight or- bits, in: Moscow Seminar on Mathematical Physics II, Amer. Math. Soc. Transl. Ser. 2, Vol. 221, Amer. Math. Soc., Providence, RI, 2007, pp. 79-120.

  22. A. Haeiger; Homotopy and integrability, in: 1971 Manifolds-Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, pp. 133-163.

  23. Хорошкин, Алгебра Ли формальных векторных полей, расширенных формальными g-знaцнымu фyнкцuямu, Зап. науч. сем. ПОМИ 335 (2006), 205-230. Engl. transl.: A. Khoroshkin, Lie algebra of formal vector fields extended by formal g-valued functions, J. Math. Sci. (N. Y.) 143 (2007), no. 1, 2816-2830.

  24. M. Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II, Paris, 1992, 97-121,

  25. R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, in: The Schur Lectures (1992) (Tel Aviv), Israel Math. Conf. Proc. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1-182.

  26. J. Humphreys, Introduction to Lie Algebras and Representation Theory, 2nd printing, revised, Graduate Texts in Mathematics 9, Springer-Verlag, New York, 1978. Russian transl.: Дж Хамфрис, Введение в теорию Алгебр Ли и ux представлений, MЦHMO, M., 2003.

  27. J. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category O, Graduate Studies in Mathematics 94, Amer. Math. Soc., Providence, RI, 2008.

  28. С. B. Лапин Дифференциальные возмущенoиa и D ∞ -дифференциальные модули,Mатем. сб. 192 (2001), вып. 11, 55-76. Engl. transl.: S. Lapin Differential perturbations and D ∞ -differential modules Sbornik: Math. 192 (2001), no. 11, 1639-1659.

  29. J. Lepowsky, A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra 49 (1977), no. 2, 496-511.

    Article  MathSciNet  MATH  Google Scholar 

  30. I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, with contributions by A. Zelevinsky, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

  31. R. Stanley, Enumerative Combinatorics, Vol. 1, 2nd edition, Cambridge Studies in Advanced Mathematics, Vol. 49. Cambridge University Press, Cambridge, 2012. Russian transl.: Р. Стeнли, Перечислительнaя комбинаторикa Mиp, M., 1990.

  32. C. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge University Press, Cambridge, 1994.

  33. H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939. Russian transl.: Г. Вейль, Классические группы, их инварианты и представления, ИЛ, M., 1947.

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Authors and Affiliations

  1. Laboratory of Mathematical Physics, National Research University, Higher School of Economics, Vavilova 7, Moscow, 101990, Russia

    A. S. KHOROSHKIN

  2. ITEP, Bolshaya Cheremushkinskaya 25, Moscow, 117259, Russia

    A. S. KHOROSHKIN

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Supported in parts by RFBR grant RFBR 15-01-09242, by The National Research University-Higher School of Economics Academic Fund Program in 2014–2015, research grant 14-01-0124, by Dynasty foundation and by Simons-IUM fellowship. The article was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.

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KHOROSHKIN, A.S. CHARACTERISTIC CLASSES OF FLAGS OF FOLIATIONS AND LIE ALGEBRA COHOMOLOGY. Transformation Groups 21, 479–518 (2016). https://doi.org/10.1007/s00031-015-9354-5

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