Advertisement

Brackets, superalgebras and spectral gap

  • Consuelo MartínezEmail author
  • Efim Zelmanov
Article
  • 6 Downloads

Abstract

The purpose of this survey is to discuss Poisson and contact brackets and related infinite dimensional superalgebras. All vector spaces are considered over the field of complex numbers \({\mathbb {C}}\).

Keywords

Bracket Superalgebra Conformal algebra 

Mathematics Subject Classification

17B68 17B63 17C90 

Notes

Acknowledgements

The first author has been partially supported by the Projects MTM 2017-83506-C2-2-P from the Ministry of Economy of Spain and FC-GRUPIN-IDI/2018/000193 from the Principado de Asturias, the second author gratefully acknowledges the support by the NSF Grant DMS 160 1920.

Author Contributions

CM, EZ designed research, performed research, and wrote the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Ademollo, M., et al.: Supersymmetric strings and color confinement. Phys. Lett. 62B, 105–110 (1976)CrossRefGoogle Scholar
  2. 2.
    Bakalov, B., D’Andrea, A., Kac, V.G.: Theory of finite pseudoalgebras. Adv. Math. 162(1), 1–140 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Billig, Y., Futorny, V.: Classification of irreducible representations of Lie algebra of vector fields on a torus. J. Reine Angew. Math. 720, 199–216 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cantarini, N., Kac, V.G.: Classification of linearly compact simple Jordan and generalized Poisson superalgebras. J. Algebra 313(1), 100–124 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cheng, S.-J., Kac, V.G.: A new \(N=6\) superconformal algebra. Commun. Math. Phys. 186(1), 219–231 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dong, C.: Vertex algebras associated with even lattices. J. Algebra 161(1), 245–265 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fattori, D., Kac, V.G.: Classification of finite simple Lie conformal superalgebras. J. Algebra 258(1), 23–59 (2002). (Special issue in celebration of Claudio Procesi’s 60th birthday)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grozman, P., Leites, D., Shchepochkina, I.: Lie superalgebras of string theories. Acta Math. Vietnam 26(1), 27–63 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kac, V.G., Martinez, C., Zelmanov, E.: Graded simple Jordan superalgebras of growth one. Mem. Am. Math. Soc. 150(711), x+140 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kac, V.G. van de Leur, J.W.: On classification of superconformal algebras. In: Strings ’88 (College Park, MD, 1988), pp. 77–106. World Scientific Publishing, Teaneck (1989)Google Scholar
  12. 12.
    Kac, V.: Vertex Algebras for Beginners, University Lecture Series, vol. 10. American Mathematical Society, Providence (1997)Google Scholar
  13. 13.
    Kantor, I.L.: Certain generalizations of Jordan algebras. Trudy Sem. Vektor. Tenzor. Anal. 16, 407–499 (1972)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kantor, I.L.: Connection Between Poisson Brackets and Jordan and Lie Superalgebras, Lie Theory, Differential Equations and Representation Theory (Montreal, PQ, 1989), pp. 213–225. University of Montréal, Montreal (1990)zbMATHGoogle Scholar
  15. 15.
    Kaplansky, I., Santharoubane, L.J.: Harish-Chandra modules over the Virasoro algebra. In: Infinite-Dimensional Groups with Applications (Berkeley, California, 1984), Mathematical Sciences Research Institute Publications, vol. 4, pp. 217–231. Springer, New York (1985)Google Scholar
  16. 16.
    Koecher, M.: Imbedding of Jordan algebras into Lie algebras. I. Am. J. Math. 89, 787–816 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lubotzky, A.: Expander graphs in pure and applied mathematics. Bull. Am. Math. Soc. (N.S.) 49(1), 113–162 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Martínez, C., Zelmanov, E.: Specializations of Jordan superalgebras. Can. Math. Bull. 45(4), 653–671 (2002). (Dedicated to Robert V. Moody)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Martínez, C., Zelmanov, E.: Representation theory of Jordan superalgebras. I. Trans. Am. Math. Soc. 362(2), 815–846 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Martí nez, C., Zelmanov, E.I.: Lie superalgebras graded by \(P(n)\) and \(Q(n)\). Proc. Natl. Acad. Sci. USA 100(14), 8130–8137 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Martin, C., Piard, A.: Classification of the indecomposable bounded admissible modules over the virasoro lie algebra with weightspaces of dimension not exceeding two. Commun. Math. Phys. 150(3), 465–493 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Martinez, C., Zelmanov, E.: Simple finite-dimensional Jordan superalgebras of prime characteristic. J. Algebra 236(2), 575–629 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mathieu, O.: Classification of Harish-Chandra modules over the Virasoro Lie algebra. Invent. Math. 107(2), 225–234 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Neveu, A., Schwarz, J.H.: Factorizable dual model of pions. Nucl. Phys. B 31(1), 86–112 (1971)CrossRefGoogle Scholar
  25. 25.
    Ramond, P.: Dual theory for free fermions. Phys. Rev. D (3) 3, 2415–2418 (1971)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schoutens, K.: A nonlinear representation of the \(d=2\) SO(4) extended superconformal algebra. Phys. Lett. B 194, 75–80 (1987)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Schwimmer, A., Seiberg, N.: Comments on the n = 2, 3, 4 superconformal algebras in two dimensions. Phys. Lett. B 184(2), 191–196 (1987)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tits, J.: Une classe d’algèbres de Lie en relation avec les algèbres de Jordan. Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24, 530–535 (1962)CrossRefzbMATHGoogle Scholar
  29. 29.
    Tits, J.: Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction. Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28, 223–237 (1966)CrossRefzbMATHGoogle Scholar

Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2019

Authors and Affiliations

  1. 1.Departmento de MatemáticasUniversidad de OviedoOviedoSpain
  2. 2.Department of MathematicsUniversity of California San DiegoLa JollaUSA

Personalised recommendations