SpringerLink
Log in

On Certain Algebraic Structures Associated with Lie (Super)Algebras

  • Conference paper
  • First Online:
Non-Associative Algebras and Related Topics (NAART 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 427))

  • 182 Accesses

Abstract

In this paper we exhibit a survey of constructions of Lie (super)algebras associated with certain triple systems, several examples and a historical story in nonassociative algebras (in particular, Jordan algebras).

Dedicated to the 60th birthday of Professor Alberto Elduque

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Allison, B.N.: A class of nonassociative algebras with involution containing the class of Jordan algebras. Math. Ann. 237(2), 133–156 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allison, B.N.: Models of isotropic simple Lie algebras. Commun. Algebr. 7(17), 1835–1875 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  3. Asano, H., Yamaguti, K.: A construction of Lie algebras by generalized Jordan triple systems of second order. Ned. Akad. Wet. (Indag. Math.) 42(3), 249–253 (1980)

    Google Scholar 

  4. Asano, H.: Classification of non-compact real simple generalized Jordan triple systems of the second kind. Hiroshima Math. J. 21, 463–489 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bertram, W.: Complex and quaternionic structures on symmetric spaces - correspondence with Freudenthal-Kantor triple systems. In: Miyaoka, R., Tamaru, H. (eds.) Theory of Lie Groups and Manifolds. Sophia Kokyuroku in Math. 45, 61–80 (2002)

    Google Scholar 

  6. Elduque, A., Kamiya, N., Okubo, S.: Simple \((-1,-1)\) balanced Freudenthal Kantor triple systems. Glas. Math. J. 11(2), 353–372 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Elduque, A., Kamiya, N., Okubo, S.: \((-1,-1)\) balanced Freudenthal Kantor triple systems and noncommutative Jordan algebras. J. Algebr. 294(1), 19–40 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Faulkner, J.R.: Structurable triples, Lie triples, and symmetric spaces. Forum Math. 6, 637–650 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Frappat, L., Sciarrino, A., Sorba, P.: Dictionary on Lie Algebras and Superalgebras. Academic Press, San Diego, California 92101–4495 (2000)

    Google Scholar 

  10. Jacobson, N.: Lie and Jordan triple systems. Am. J. Math. 71, 149–170 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jacobson, N.: Structure and representations of Jordan algebras. Am. Math. Soc. Colloq. Publ. XXXIX. American Mathematical Society, Providence (1968)

    Google Scholar 

  12. Kac, V.G.: Lie superalgebras. Adv. Math. 26(1), 8–96 (1977)

    Google Scholar 

  13. Kamiya, N.: A structure theory of Freudenthal-Kantor triple systems. J. Algebr. 110(1), 108–123 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kamiya, N.: A construction of anti-Lie triple systems from a class of triple systems. Mem. Fac. Sci. Shimane Univ. 22, 51–62 (1988)

    MathSciNet  MATH  Google Scholar 

  15. Kamiya, N.: A structure theory of Freudenthal-Kantor triple systems II. Comment. Math. Univ. St. Paul. 38(1), 41–60 (1989)

    MathSciNet  MATH  Google Scholar 

  16. Kamiya, N.: A structure theory of Freudenthal-Kantor triple systems III. Mem. Fac. Sci. Shimane Univ. 23, 33–51 (1989)

    MathSciNet  MATH  Google Scholar 

  17. Kamiya, N.: On \((\epsilon ,\delta )\)-Freudenthal-Kantor triple systems. Nonassociative algebras and related topics (the conference of Hiroshima, 1990), pp. 65–75. World Scientific Publishing, River Edge (1991)

    Google Scholar 

  18. Kamiya, N.: The construction of all simple Lie algebras over \(\bf {C}\) from balanced Freudenthal-Kantor triple systems. Contrib. Gen. Algebr. 7 (Vienna, 1990), 205–213, Hölder-Pichler-Tempsky, Vienna (1991)

    Google Scholar 

  19. Kamiya, N.: On Freudenthal-Kantor triple systems and generalized structurable algebras. Non-associative algebra and its applications (Oviedo, 1993), pp. 198–203. Math. Appl. 303. Kluwer Academic Publishers, Dordrecht (1994)

    Google Scholar 

  20. Kamiya, N.: On the Peirce decompositions for Freudenthal-Kantor triple systems. Commun. Algebr. 25(6), 1833–1844 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kamiya, N.: On a realization of the exceptional simple graded Lie algebras of the second kind and Freudenthal-Kantor triple systems. Bull. Pol. Acad. Sci. Math. 46(1), 55–65 (1998)

    MathSciNet  MATH  Google Scholar 

  22. Kamiya, N., Okubo, S.: On \(\delta \)-Lie supertriple systems associated with \((\epsilon ,\delta )-\) Freudenthal-Kantor supertriple systems. Proc. Edinb. Math. Soc. 43(2), 243–260 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kamiya, N., Okubo, S.: A construction of Jordan superalgebras from Jordan-Lie triple systems. In: Costa, Peresi, etc. (eds.) Lecture Notes in Pure and Applied Mathematics, vol. 211. Non-Associative Algebra and Its Applications, pp. 171–176. Marcel Dekker Inc (2002)

    Google Scholar 

  24. Kamiya, N., Okubo, S.: A construction of simple Jordan superalgebra of \(F\) type from a Jordan-Lie triple system. Ann. Mate. Pura Appl. 181, 339–348 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kamiya, N., Okubo, S.: Construction of Lie superalgebras \(D(2,1;\alpha ), G(3)\) and \(F(4)\) from some triple systems. Proc. Edinb. Math. Soc. 46(1), 87–98 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kamiya, N., Okubo, S.: On generalized Freudenthal-Kantor triple systems and Yang-Baxter equations. In: Proceedings of the XXIV International Colloquium Group Theoretical Methods in Physics, IPCS, vol. 173, pp. 815–818 (2003)

    Google Scholar 

  27. Kamiya, N., Okubo, S.: A construction of simple Lie superalgebras of certain types from triple systems. Bull. Aust. Math. Soc. 69(1), 113–123 (2004)

    Article  MATH  Google Scholar 

  28. Kamiya, N.: Examples of Peirce decomposition of generalized Jordan triple system of second order-Balanced cases. In: Fuchs , J. (ed.) Noncommutative Geometry and Representation Theory in Mathematical Physics, pp. 157–165. Contemp. Math. 391. AMS, Providence, RI (2005)

    Google Scholar 

  29. Kamiya, N., Okubo, S.: Composition, quadratic, and some triple systems. Non-associative algebra and its applications. Lecture Notes in Pure and Applied Mathematics, vol. 246, pp. 205–231. Chapman & Hall/CRC, Boca Raton (2006)

    Google Scholar 

  30. Kamiya, N., Mondoc, D.: A new class of nonassociative algebras with involution. Proc. Jpn. Acad., Ser. A 84(5), 68–72 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kamiya, N., Mondoc, D., Okubo, S.: A structure theory of \((-1,-1)\)-Freudenthal Kantor triple systems. Bull. Aust. Math. Soc. 81, 132–155 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kamiya, N., Mondoc, D., Okubo, S.: A characterization of (-1,-1)-Freudenthal-Kantor triple systems. Glas. Math. J. 53, 727–738 (2011)

    Google Scholar 

  33. Kamiya, N., Mondoc, D., Okubo, S.: A review of Peirce decomposition for unitary (-1,-1)-Freudenthal-Kantor triple systems. In: Makhlouf, A., Paal, E., Silvestarov, S., Stolin, A. (eds.) Proceedings in Mathematics and Stastics, vol. 85, pp. 145–155. Springer (2014)

    Google Scholar 

  34. Kamiya, N., Okubo, S.: On triality of structurable and pre-structurable algebras. J. Algebr. 416, 58–88 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kamiya, N., Okubo, S.: Algebras and groups satisfying triality relations monograph(book). University of Aizu (2015). Algebras, Groups and Geometries, vol. 33, pp. 1–92 (2016) and Algebras, Groups and Geometries, vol. 35, pp. 113–168 (2018)

    Google Scholar 

  36. Kamiya, N., Okubo, S.: Symmetry of Lie algebras associated with \((\epsilon .\delta )\) Freudenthal-Kantor triple systems. Proc. Edinb. Math. Soc. 89, 169–192 (2016)

    Google Scholar 

  37. Kamiya, N., Sato, M.: Hermitian \((\varepsilon ,\delta )\)-Freudenthal-Kantor triple systems and certain applications of \(^{*}\)-Generalized Jordan triple systems to field theory. Adv. High Energy Phys. (2014). https://doi.org/10.1155/2014/310264

  38. Kamiya, N., Sato, M.: Hermitian triple systems associated with bilinear forms and certain applications to field theory. Hadron. J. 37(2), 131–147 (2014)

    Google Scholar 

  39. Kamiya, N., Shibukawa, Y.: Dynamical Yang-Baxter maps and weak Hopf algebras associated with quandles. In: Yamane, Kogiso, Koga, Kimura (eds.) Proceedings of the Meeting for Study of Number theory, Hopf Algebras and Related topics, pp. 1–23. Yokohama Publishers (2019)

    Google Scholar 

  40. Kaneyuki, S., Asano, H.: Graded Lie algebras and generalized Jordan triple systems. Nagoya Math. J. 112, 81–115 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kantor, I.L.: Graded Lie algebras. Tr. Sem. Vect. Tens. Anal. 15, 227–266 (1970)

    MATH  Google Scholar 

  42. Kantor, I.L.: Some generalizations of Jordan algebras. Tr. Sem. Vect. Tens. Anal. 16, 407–499 (1972)

    MATH  Google Scholar 

  43. Kantor, I.L.: Models of exceptional Lie algebras. Sov. Math.-Dokl. 14(1), 254–258 (1973)

    MATH  Google Scholar 

  44. Kantor, I.L.: A generalization of the Jordan approach to symmetric Riemannian spaces. The Monster and Lie Algebras (Columbus, OH, 1996), pp. 221–234. Ohio State University Mathematics Research Institute Publications, vol. 7. de Gruyter, Berlin (1998)

    Google Scholar 

  45. Kantor, I.L., Kamiya, N.: A Peirce decomposition for generalized Jordan triple systems of second order. Commun. Algebr. 31(12), 5875–5913 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Loos, O.: Symmetric Spaces. I: General Theory. W. A. Benjamin, Inc., New York-Amsterdam (1969)

    Google Scholar 

  47. Koecher, M.: Embedding of Jordan algebras into Lie algebras I. II. Am. J. Math. 89, 787–16 (1967) and 90, 476–510 (1968)

    Google Scholar 

  48. Lister, W.G.: A structure theory of Lie triple systems. Trans. Am. Math. Soc. 72, 217–242 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  49. Meyberg, K.: Lectures on algebras and triple systems. Lecture Notes. The University of Virginia, Charlottesville (1972)

    Google Scholar 

  50. Mondoc, D.: Models of compact simple Kantor triple systems defined on a class of structurable algebras of skew-dimension one. Commun. Algebr. 34(10), 3801–3815 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  51. Mondoc, D.: On compact realifications of exceptional simple Kantor triple systems. J. Gen. Lie Theory Appl. 1(1), 29–40 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Mondoc, D.: Compact realifications of exceptional simple Kantor triple systems defined on tensor products of composition algebras. J. Algebr. 307(2), 917–929 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  53. Mondoc, D.: Compact exceptional simple Kantor triple systems defined on tensor products of composition algebras. Commun. Algebr. 35(11), 3699–3712 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. Neher, E.: Jordan triple systems by the grid approach. Lecture Notes in Mathematics, vol. 1280. Springer, Berlin (1987)

    Google Scholar 

  55. Okubo, S.: Introduction to octonion and other non-associative algebras in physics. Montroll Memorial Lecture Series in Mathematical Physics, vol. 2. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  56. Okubo, S., Kamiya, N.: Jordan-Lie superalgebra and Jordan-Lie triple system. J. Algebr. 198(2), 388–411 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  57. Okubo, S., Kamiya, N.: Quasi-classical Lie superalgebras and Lie supertriple systems. Commun. Algebr. 30(8), 3825–3850 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  58. Okubo, S.: Symmetric triality relations and structurable algebras. Linear Algebr. Appl. 396, 189–222 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  59. Satake, I.: Algebraic Structures of Symmetric Domains. Princeton University (1980)

    Google Scholar 

  60. Scheunert, M.: The theory of Lie superalgebras. An introduction. Lecture Notes in Mathematics, vol. 716. Springer, Berlin (1979)

    Google Scholar 

  61. Springer, T.: Jordan Algebras and Algebraic Groups. Springer (1973)

    Google Scholar 

  62. Tits, J.: Alg\(\grave{\rm e}\)bres alternatives, alg\(\grave{\rm e}\)bres de Jordan, et alg\(\grave{\rm e}\)bres de Lie exceptionnelles. Ned. Acad. Wet. Proc. Ser. A 69, 223–237 (1966)

    Google Scholar 

  63. Yamaguti, K., Ono, A.: On representations of Freudenthal-Kantor triple systems \(U(\epsilon ,\delta )\). Bull. Fac. Sch. Educ. Hiroshima Univ. 7, no. II, 43–51 (1984)

    Google Scholar 

  64. Zhevlakov, K.A., Slinko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are Nearly Associative. Academic Press Inc, New York-London (1982)

    Google Scholar 

  65. Zelmanov, E.: Primary Jordan triple systems. Sib. Mat. Zh. 4, 23–37 (1983)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank the organizers of NAART II, in particular Prof. Dr. Jose Brox for exchanging several email messages. Due to COVID-19 and the travel restrictions imposed by the government of Japan, the author was unable to travel to Portugal to participate in the NAART II Conference dedicated to honor Alberto Elduque on the occasion of his 60th birthday, but he is grateful to have been able to contribute to the book dedicated to him. The author is also grateful for the referee’s comments and suggestions.

Author information

Authors and Affiliations

  1. University of Aizu, Aizuwakamatsu, 965-8580, Japan

    Noriaki Kamiya

Authors
  1. Noriaki Kamiya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Noriaki Kamiya .

Editor information

Editors and Affiliations

  1. Mathematics, University of Coimbra, Coimbra, Portugal

    Helena Albuquerque

  2. Department of Mathematics, Centre for Mathematics of the University of Coimbra, Coimbra, Portugal

    Jose Brox

  3. Mathematics, University of Oviedo, Oviedo, Spain

    Consuelo Martínez

  4. CMUC-Centre for Mathematics of the University of Coimbra, CeBER-Centre for Business and Economics Research, Faculty of Economics, University of Coimbra, Coimbra, Portugal

    Paulo Saraiva

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kamiya, N. (2023). On Certain Algebraic Structures Associated with Lie (Super)Algebras. In: Albuquerque, H., Brox, J., Martínez, C., Saraiva, P. (eds) Non-Associative Algebras and Related Topics. NAART 2020. Springer Proceedings in Mathematics & Statistics, vol 427. Springer, Cham. https://doi.org/10.1007/978-3-031-32707-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation