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Structures of G(2) Type and Nonintegrable Distributions in Characteristic p

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Abstract

Lately, the following were observed: (1) an upsurge of interest (in particular, triggered by a paper by Atiyah and Witten) to manifolds with G(2)-type structure; (2) classifications are obtained of simple (finite dimensional and graded vectorial) Lie superalgebras over fields of complex and real numbers and of simple finite dimensional Lie algebras over algebraically closed fields of characteristic p>3. The importance of nonintegrable distributions in the above classifications were observed and illustrated by an explicit description of several exceptional simple Lie algebras for p=2, 3 (Melikyan algebras; Brown, Ermolaev, Frank, and Skryabin algebras) as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4) and Sp(10). The description is performed in terms of Cartan–Tanaka–Shchepochkina prolongs and is similar to descriptions of simple Lie superalgebras of vector fields with polynomial coefficients. Our results illustrate usefulness of Shchepochkina’s algorithm and SuperLie package; two families of simple Lie algebras found in the process are new; one more, rather mysterious, is partly elucidated

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References

  1. Adams J.F. (1996). Lectures on exceptional Lie groups. University of Chicago Press, Chicago, IL, xiv+122 pp.

    MATH  Google Scholar 

  2. Atiyah M., Witten E. (2002). M-theory dynamics on a manifold of G 2 holonomy. Adv. Theor. Math. Phys. 6(1):1–106, MR1992874 (2004f:53046)

    MATH  MathSciNet  Google Scholar 

  3. Benkart G., Elduque A. (2003). The Tits construction and the exceptional simple classical Lie superalgebras. Q. J. Math. 54(2):123–137

    Article  MATH  MathSciNet  Google Scholar 

  4. Benkart G., Kostrikin A.I., Kuznetsov M.I. (1996). The simple graded Lie algebras of characteristic three with classical reductive component L 0. Comm. Algebra 24(1):223–234

    Article  MATH  MathSciNet  Google Scholar 

  5. Brown G. (1995). Families of simple Lie algebras of characteristic two. Comm. Algebra 23(3):941–954

    Article  MATH  MathSciNet  Google Scholar 

  6. Bryant, R.L.: Some remarks on G 2-structures; math.DG/0305124

  7. Cartan, É.: Über die einfachen Transformationsgrouppen, Leipziger Berichte (1893), 395–420. Reprinted in: \OE uvres complètes. Partie II. (French) [Complete works. Part II] Algèbre, systèmes différentiels et problèmes d’équivalence. [Algebra, differential systems and problems of equivalence] Second edition. Éditions du Centre National de la Recherche Scientifique (CNRS), Paris (1984)

  8. Cremmer E., Julia B. (1979). The SO(8) supergravity. Nuclear Phys. B 159(1–2):141–212

    Article  ADS  MathSciNet  Google Scholar 

  9. Ermolaev, Yu. B.: Integral bases of classical Lie algebras. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. no. 3, 16–25 (2004); translation in Russian Math. (Iz. VUZ) 48(3) 13–22 (2004)

  10. Fei Q.-Y., Shen G.-Y. (1992). Universal graded Lie algebras, J. Algebra 152:439–453

    Article  MATH  MathSciNet  Google Scholar 

  11. Fernando S.L. (1990). Lie algebra modules with finite-dimensional weight spaces. Trans. Amer. Math. Soc. 332:757–781

    Article  MathSciNet  Google Scholar 

  12. Fock, V.V., Goncharov, A.B.: Cluster X-varieties, amalgamation and Poisson-Lie groups. math.RT/0508408

  13. Fuks, (Fuchs) D.: Cohomology of infinite dimensional Lie algebras. Consultants Bureau, NY (1986)

  14. Grozman, P.: Classification of bilinear invariant operators on tensor fields. Functional Anal. Appl. 14(2), 127–128 (1980); for details, see id., ESI-preprint 1114 (2001) (http://www.esi.ac.at); math.RT/0509562

  15. Grozman, P.: SuperLie, http://www.equaonline.com/math/SuperLie

  16. Grozman P., Leites D. (2001). Defining relations for classical Lie superalgebras with Cartan matrix. Czech. J. Phys. 51(1):1–22 hep-th/9702073

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Grozman, P., Leites, D.: SuperLie and problems (to be) solved with it. Preprint MPIM-Bonn, 2003-39 (http://www.mpim-bonn.mpg.de)

  18. Grozman, P., Leites, D.: The nonholonomic Riemann and Weyl tensors for flag manifolds. math.DG/0509399

  19. Grozman, P., Leites, D.: Structures of G(2) type and nonintegrable distributions in characteristic p; math.RT/0509400

  20. Grozman, P.: Leites, D.: Lie superalgebras of supermatrices of complex size. Their generalizations and related integrable systems. In: Ramírez de Arellano, E., Shapiro, M., Tovar, L., Vasilevski, N. (eds.) Proceedings of International symposium complex analysis and related topics, Mexico, 1996, Birkhäuser, pp. 73–105; math.RT/0202177 (1999)

  21. Grozman P., Leites D., Poletaeva E. (2002). Defining relations for classical Lie superalgebras without Cartan matrices). Homology, Homotopy Appl. 4(2):259–275 math.RT/0202152

    MATH  MathSciNet  Google Scholar 

  22. Grozman, P., Leites, D., Shchepochkina, I.: Invariant operators on supermanifolds and standard models. In: Olshanetsky, M., Vainstein, A. (eds.) Multiple facets of quantization and supersymmetry. Michael Marinov Memorial Volume, World Scientific, River Edge, NJ, pp. 508–555. [ESI preprint 1111 (2001)(http://www.esi.ac.at); math.RT/0202193] (2002)

  23. Holmes R.R. (2001). Simple modules with character height at most one for the restricted Witt algebras. J. Algebra 237(2):446–469

    Article  MATH  MathSciNet  Google Scholar 

  24. Kantor, I.L.: Infinite dimensional simple graded Lie algebras. (Russian) Dokl. Akad. Nauk SSSR 179 1968 534–537. English translation: Soviet Math. Dokl. 9, 409–412 (1968)

  25. Kochetkov, Y., Leites, D.: Simple Lie algebras in characteristic 2 recovered from superalgebras and on the notion of a simple finite group. In: Proceedings of the international conference on algebra, part 2 (Novosibirsk, 1989), Contemp. Math., 131, Part 2, Amer. Math. Soc., Providence, RI, pp. 59–67 (1992)

  26. Kostrikin, A.I.: A parametric family of simple Lie algebras. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 34 744–756; English translation: Math. USSR-Izv. 4, 751–764 (1970)

  27. Kuznetsov M.I. (2001). Graded Lie algebras with the almost simple component L 0. Pontryagin Conference, 8, Algebra (Moscow, 1998). J. Math. Sci. (New York) 106(4):3187–3211

    Article  MathSciNet  Google Scholar 

  28. Kuznetsov M.I. (1991). The Melikyan algebras as Lie algebras of the type G 2. Comm. Algebra 19(4): 1281–1312

    Article  MATH  MathSciNet  Google Scholar 

  29. Larsson, T.: Structures preserved by exceptional Lie algebras; math-ph/0301006

  30. Lebedev, A.: Nondegenerate bilinear forms in characteristic 2, related contact structures and simple Lie algebras (to appear)

  31. Leites D., Shchepochkina I. (2001). How to quantize the antibracket. Theor. and Math. Physics 126(3): 339–369 preprint ESI-875 (2001) (http://www.esi.ac.at); math-ph/0510048

    Article  MathSciNet  Google Scholar 

  32. Leites, D., Shchepochkina, I.: Classification of the simple Lie superalgebras of vector fields, preprint MPIM-2003-28 (http://www.mpim-bonn.mpg.de)

  33. Mathieu O. (2000). Classification of irreducible weight modules. Ann. Inst. Fourier (Grenoble) 50:537–592

    MATH  MathSciNet  Google Scholar 

  34. Melikyan, G.: On simple Lie algebras of characteristic 5, Uspekhi Mat. Nauk 35, 203–204; transl. Russian Math. Surveys 35, 219–220. MR0565576 (81d:17011) (1980)

  35. Rittenberg V., Scheunert M. (1982). A remarkable connection between the representations of the Lie superalgebras \(\mathfrak{osp} (1, 2n)\) and the Lie algebras \(\mathfrak{o} (2n+1)\). Commun. Math. Phys., 83:1–9

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Rudakov, A.N., Shafarevich, I.R.: Irreducible representations of a simple three-dimensional Lie algebra over a field of finite characteristic. (Russian) Mat. Zametki 2, 439–454 (1967) (1967, 2, 760–767) MR0219583 (36 #2662)

  37. Shchepochkina I. Five exceptional simple Lie superalgebras of vector fields and their fourteen regradings. Representation Theory (electronic journal of AMS) 3:373–415 hep-th/9702121

  38. Shchepochkina, I.: How to realize Lie algebras by vector fields; math.RT/0509472

  39. Shchepochkina, I.: How to realize Lie algebras by vector fields. Examples (to appear)

  40. Shchepochkina, I.: Maximal subalgebras of simple vectorial Lie superalgebras (to appear)

  41. Shen G.-Y. (2002). Lie algebras of CL type. J. Algebra 249(1):95–109

    Article  MATH  MathSciNet  Google Scholar 

  42. Sergeev, A.: Orthogonal polynomials and Lie superalgebras; math.RT/9810110

  43. Skryabin, S.M.: New series of simple Lie algebras of characteristic 3. (Russian) Mat. Sb. 183(8), 3–22 (1992); translation in Russian Acad. Sci. Sb. Math. 76(2), 389–406 (1993)

  44. Strade, H.: Simple Lie algebras over fields of positive characteristic. I. Structure theory. de Gruyter Expositions in Mathematics, 38. Walter de Gruyter, Berlin, pp. viii+540 (2004)

  45. Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. Progress in differential geometry, Adv. Stud. Pure Math., 22, Math. Soc. Japan, Tokyo, pp. 413–494 (1993)

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Author notes

  1. Dimitry Leites

    Present address: MPIMIS, Inselstr. 22, DE-04103, Leipzig, Germany

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  1. Equa Simulation AB, Stockholm, Sweden

    Pavel Grozman

  2. Department of Mathematics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, SE-104 05, Stockholm, Sweden

    Dimitry Leites

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  1. Pavel Grozman
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  2. Dimitry Leites
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Correspondence to Dimitry Leites.

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In memory of Felix Aleksandrovich Berezin

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Grozman, P., Leites, D. Structures of G(2) Type and Nonintegrable Distributions in Characteristic p . Lett Math Phys 74, 229–262 (2005). https://doi.org/10.1007/s11005-005-0026-6

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