Skip to main content

About Nonassociativity in Mathematics and Physics

Abstract

A short review about nonassociative algebraic systems (mainly nonassociative algebras) and their physical applications is presented. We begin with some motivations, then we give a brief historical overview about the formation and development of the concept of hypercomplex number system and about some earlier applications. The main directions discussed are the octonionic, Lie-admissible, and quasigroup approaches. Also, some problems investigated in Tartu, the octonionic approach, Moufang–Mal'tsev symmetry, and associator quantization are discussed. This review does not pretend to be complete as the accent is placed on ideas and not on the techniques, also the references are quite sporadic (there are many authors and results mentioned in the text without references).

This is a preview of subscription content, access via your institution.

References

  1. Akivis, M. A.: About geodesic loops and local ternary systems of the affinely connected spaces, Sibirsk. Mat. Zh. 19 (1978), 243-253 (in Russian).

    Google Scholar 

  2. Albert, A. A.: Structure of Algebras, Amer. Math. Soc., Providence, RI, 1939.

    Google Scholar 

  3. Albert, A. A.: Power-associativity of rings, Trans. Amer. Math. Soc. 64 (1948), 552-593.

    Google Scholar 

  4. Asano, H.: On a class of nonassociative algebras, Yokohama Math. J. 20 (1972), 143-149.

    Google Scholar 

  5. Batalin, I. A.: Quasigroup construction and first class constraints, J. Math. Phys. 22 (1981), 1837-1856.

    Google Scholar 

  6. Batalin, I. A. and Vilkovisky, G. A.: Existence theorem for gauge algebra, J. Math. Phys. 27 (1985), 172-184.

    Google Scholar 

  7. Benkart, G. M. and Osborn, J. M.: Flexible Lie-admissible algebras, J. Algebra 7 (1981), 11-31.

    Google Scholar 

  8. Benkart, G. M.: Power-associative Lie-admissible algebras, J. Algebra 90 (1984), 37-58.

    Google Scholar 

  9. Berezin, A. V., Kurochkin, Yu. A. and Tolkachev, E. A.: Quaternions in Relativistic Physics, Nauka i Tehnika, Minsk (Belarus), 1989 (in Russian).

    Google Scholar 

  10. Birkhoff, G. and von Neumann, J.: The logic of quantum mechanics, Ann. of Math. (USA) 37 (1936), 823-843.

    Google Scholar 

  11. Bohm, A., Ne'eman, Y. and Barut, A. O.: Dynamical Groups and Spectrum-Generating Algebras I, II, World Scientific, Singapore, 1988.

    Google Scholar 

  12. Braun, H. and Koecher, M.: Jordan-Algebren, Springer, New York, 1966.

    Google Scholar 

  13. Bruck, R. H.: A Survey of Binary Systems, Springer, New York, 1958.

    Google Scholar 

  14. Cayley, A.: On Jacobi's elliptic functions, in reply to the Rev. Brice Bronwin; and on quaternions, Philos. Mag. (3) 26 (1845), 210-213.

    Google Scholar 

  15. Conway, J. H. and Sloane, N. J. A.: Sphere Packings, Lattices and Groups, Springer, New York, 1988.

    Google Scholar 

  16. Dickson, L. E.: On quaternions and their generalization and the history of the eight square theorem, Ann. of Math. (USA) 20 (1919), 155-171.

    Google Scholar 

  17. Duff, M. J., Nilsson, B. E. W. and Pope, C. N.: Kaluza-Klein supergravity, Phys. Rep. 130 (1986), 1-142.

    Google Scholar 

  18. Dyson, F. J.: Mathematics in the physical sciences, Sci. Amer. 211 (1964), 128-146.

    Google Scholar 

  19. Eilenberg, S.: Extension of general algebras, Ann. Soc. Polon. Math. 21 (1948), 125-134.

    Google Scholar 

  20. Gell-Mann, M. and Ne'eman, Y. (eds.): The Eightfold Way, Benjamin, New York, 1964.

    Google Scholar 

  21. Goldstine, H. H. and Horwitz, L. P.: Hilbert space with nonassociative scalars I, II, Math. of Ann. 154 (1964), 1-27; 164 (1966), 291-316.

    Google Scholar 

  22. Graves, J. T.: On a connection between the general theory of normal couples and the theory of complete quadratic functions of two variables, Philos. Mag. (3) 26 (1845), 315-320.

    Google Scholar 

  23. Green, M. B., Schwarz, J. H. and Witten, E.: Superstring Theory I, II, Cambridge Univ. Press, 1987.

  24. Günaydin, M. and Gürsey, F.: Quark structure and octonions, J. Math. Phys. 14 (1973), 1615-1667.

    Google Scholar 

  25. Günaydin, M., Piron, C. and Ruegg, H.: Moufang plane and octonionic quantum mechanics, Comm. Math. Phys. 61 (1978), 69-85.

    Google Scholar 

  26. Gustafson, W. H.: The history of algebras and their representations, in Lecture Notes in Math. 944, Springer, New York, 1982, pp. 1-28.

    Google Scholar 

  27. Houtappel, R. M. F., van Dam, H. and Wigner, E.: The conceptual basis and use of the geometric invariance principles, Rev. Modern Phys. 37 (1965), 595-632.

    Google Scholar 

  28. Jacobson, N.: Lie and Jordan triple systems, Amer. J. Math. 71 (1949), 149-170.

    Google Scholar 

  29. Jacobson, N.: Structure and Representations of Jordan Algebras, Amer. Math. Soc., Providence, RI, 1968.

    Google Scholar 

  30. Jauch, J. M.: Foundations of Quantum Mechanics, Addison-Wesley, Reading, Mass., 1968.

    Google Scholar 

  31. Jordan, P.: Über eine Klasse nichtassoziativer hyperkomplexer Algebren, Göttingen Nachr. (1932), 569-575.

  32. Jordan, P.: Über die Multiplikation quantenmechanischen Grossen, Z. Phys. 80 (1933), 285-291.

    Google Scholar 

  33. Jordan, P.: Über Verallgemeinerungs-möglichkeiten des Formalismus der Quantenmechanik, Göttingen Nachr. (1933), 209-217.

  34. Jordan, P.: Über das verhältnis der Theorie der Elementarlange zur Quantentheorie I, II, Comm. Math. Phys. 9 (1968), 279-292; 11 (1968), 293-296.

    Google Scholar 

  35. Jordan, P.: Zur Frage einer physikalischen Verwendbarkeit nichtassoziativer Algebren, Z. Phys. 229 (1969), 193-198.

    Google Scholar 

  36. Jordan, P., von Neumann, J. and Wigner, E.: On an algebraic generalization of the quantum mechanical formalism, Ann. of Math. (USA) 35 (1934), 29-64.

    Google Scholar 

  37. Kadeishvili, J. D.: Santilli's Isotopies of Contemporary Algebras, Geometries and Relativities, Hadronic Press, Palm Harbor (Florida, USA), 1992.

    Google Scholar 

  38. Klein, F.: Vergleichende Betrachtungen über neuere geometrische Forschungen, Program zu Eintritt in die philosophische Facultät und den Senat der Universität zu Erlangen, Deichert, Erlangen, 1872.

  39. Koecher, M.: Was sind und was sollen Algebren, Math. Phys. Semesterber. 23 (1976), 174-191.

    Google Scholar 

  40. Kuusk, P. and Paal, E.: Geodesic multiplication as a tool for classical and quantum gravity, Trans. Tallinn Tech. Univ. 733 (1992), 33-42.

    Google Scholar 

  41. Kuz'min, E. N.: Mal'tsev algebras an their representations, Algebra i Logika 7 (1968), 48-69.

    Google Scholar 

  42. Langacker, P.: Grand unified theories and proton decay, Phys. Rep. 72 (1981), 185-385.

    Google Scholar 

  43. Lister, W. G.: Ternary rings, Trans. Amer. Math. Soc. 154 (1971), 37-65.

    Google Scholar 

  44. Lõhmus, J., Paal, E., and Sorgsepp, L.: On currents and symmetries associated with Mal'tsev algebras, Preprint F-50, Acad. Sci. Estonian SSR, Sect. Phys. & Astronomy, Tartu, 1989.

    Google Scholar 

  45. Lõhmus, J., Paal, E., and Sorgsepp, L.: Nonassociativity in mathematics and physics, in J. Lõhmus and P. Kuusk (eds), Quasigroups and Nonassociative Algebras in Physics, Trans. Inst. Phys. Estonian Acad. Sci. 66, Tartu, 1990, pp. 8-22 (in Russian).

  46. Lõhmus, J., Paal, E., and Sorgsepp, L.: Moufang symmetries and conservation laws, Proc. Estonian Acad. Sci., Phys.-Math. 41 (1992), 133-141.

    Google Scholar 

  47. Lõhmus, J. and Sorgsepp, L.: About nonassociative extension of matrix structure of Dirac equation, in Group-Theoretical Methods in Physics, Proc. 3rd Internat. Sem. Yurmala, Riga, Latvia, 22-24 May, 1985, vol. 2, Nauka, Moscow, 1986, pp. 603-608.

    Google Scholar 

  48. Lõhmus, J. and Sorgsepp, L.: About the hypercomplex formulation of the self-duality condition in dimensions 4 and 8, in I. Ots and V. Rosenhaus (eds), Fundamental Fields, Trans. Inst. Phys. Estonian Acad. Sci. 64, Tartu, 1989, pp. 125-139.

  49. Lõhmus, J. and Sorgsepp, L.: Aspects of self-duality in hypercomplex formalism, in J. Lõhmus and P. Kuusk (eds), Quasigroups and Nonassociative Algebras in Physics, Trans. Inst. Phys. Estonian Acad. Sci. 66, Tartu, 1990, pp. 179-197.

  50. Lõhmus, J. and Sorgsepp, L.: Ternary algebra of sedenions, in J. Lõhmus and P. Kuusk (eds), Quasigroups and Nonassociative Algebras in Physics, Trans. Inst. Phys. Estonian Acad. Sci. 66, Tartu, 1990, pp. 169-178.

  51. Lõhmus, J. and Sorgsepp, L.: Nonassociativity as a fundamental principle, Ann. Estonian Phys. Soc. (1991), 47-57 (in Estonian).

  52. Lõhmus, J. and Sorgsepp, L.: About associator quantization, Ann. Estonian Phys. Soc. (1992), 70-76 (in Estonian).

  53. MacFarlane, A.: Bibliography of Quaternions and Allied Systems of Mathematics, Dublin, 1904.

  54. Mal'tsev, A. I.: Analytical loops, Matem. Sb. 36 (1955), 569-576 (in Russian).

  55. McCrimmon, K.: Jordan algebras and their applications, Bull. Amer. Math. Soc. 84 (1978), 612-617.

    Google Scholar 

  56. McCrimmon, K.: The Russian revolution in Jordan algebras, Algebras, Groups Geom. 1 (1984), 1-61.

    Google Scholar 

  57. Molien, T.: Ueber Süsteme höherer complexer Zahlen, Doctordissertation, Universität Dorpat (Tartu), 1892.

  58. Moufang, R.: Zur Struktur von Alternativkörper, Math. Ann. 110 (1934), 416-430.

    Google Scholar 

  59. Myung, H.-C.: Lie Algebras and Flexible Lie-Admissible Algebras, Hadronic Press, Nonantum, 1982.

  60. Myung, H.-C.: A Malcev-admissible mutation of an alternative algebra, Bull. Korean Math. Soc. 20, 37-43.

  61. Myung, H.-C.: Malcev-Admissible Algebras, Birkhäuser, Basel, 1986.

    Google Scholar 

  62. Neher, E.: On the classification of Lie and Jordan triple systems, Comm. Algebra 13 (1985), 2615-2667.

    Google Scholar 

  63. Nesterov, A. I.: Quasigroup ideas in physics, in J. Lõhmus and P. Kuusk (eds), Quasigroups and Nonassociative Algebras in Physics, Trans. Inst. Phys. Estonian Acad. Sci. 66, Tartu, 1990, pp. 107-120.

  64. Osborn, J. M.: What are nonassociative algebras?, Algebras, Groups Geom. 3 (1986), 264-285.

    Google Scholar 

  65. Paal, E.: An introduction to the Moufang-symmetry, Preprint F-42, Institute of Physics, Acad. Sci. Estonian SSR, Tartu, 1987 (in Russian).

    Google Scholar 

  66. Paal, E.: Moufang-transformations, in P. Kuusk and J. Lõhmus (eds), Fundamental Interactions, Trans. Inst. Phys. Acad. Sci. Estonian SSR 62, Tartu, 1987, pp. 142-158 (in Russian).

  67. Paal, E.: Analytic Moufang-transformations, Preprint F-46, Institute of Physics, Acad. Sci. Estonian SSR, Tartu, 1988.

    Google Scholar 

  68. Paal, E.: About bicombinatorial representation of Moufang loops, in I. Ots and V. Rosenhaus (eds), Fundamental Fields, Trans. Inst. Phys. Estonian Acad. Sci. 64, Tartu, 1989, pp. 104- 124.

  69. Paal, E.: Birepresentations of derivative Moufang loops, Proc. Estonian Acad. Sci., Phys.-Math. 40 (1991), 105-111.

    Google Scholar 

  70. Paal, E.: Moufang-Mal'tsev symmetry, Proc. Estonian Acad. Sci., Phys.-Math. 42 (1993), 157-165.

    Google Scholar 

  71. Peirce, B.: Linear associative algebra, Amer. J. Math. 4 (1881), 97-229.

    Google Scholar 

  72. Piron, C.: Foundations of Quantum Physics, Addison-Wesley, Reading, Mass., 1976.

    Google Scholar 

  73. Ruegg, H.: Octonionic quark confinement, Acta Phys. Polon. B 9 (1978), 1037-1050.

    Google Scholar 

  74. Sabinin, L. V.: Odules as a new approach to the geometry with connection, Dokl. Akad. Nauk SSSR 233 (1977), 800-803 (in Russian).

    Google Scholar 

  75. Sabinin, L. V.: Methods of nonassociative algebra in differential geometry, Addendum to the Russian translation of S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Nauka, Moscow, 1981, vol. 1, pp. 293-339.

    Google Scholar 

  76. Sabinin, L. V.: Quasigroups, geometry, and physics, in J. Lõhmus and P. Kuusk (eds), Quasigroups and Nonassociative Algebras in Physics, Trans. Inst. Phys. Estonian Acad. Sci. 66, Tartu, 1990, pp. 24-54.

  77. Sakharov, A. D.: Vacuum quantum fluctuations in curved space and the theory of gravitation, Dokl. Akad. Nauk SSSR 177 (1967), 70-71 (in Russian).

    Google Scholar 

  78. Santilli, R. M.: Imbedding of Lie algebras in nonassociative structures, Nuovo Cimento A 51 (1967), 570-576.

    Google Scholar 

  79. Santilli, R. M.: An introduction to Lie-admissible algebras, Nuovo Cimento, Suppl. 6 (1968), 1225-1249.

    Google Scholar 

  80. Santilli, R. M.: Isotopic Generalizations of Galilei and Einstein's Relativities. I. Mathematical Foundations; II. Classical Isotopies, Hadronic Press, Palm Harbor, 1981, 1991.

    Google Scholar 

  81. Santilli, R. M.: Foundations of Theoretical Mechanics. I. The Inverse Problem in Newtonian Mechanics; II. Hamiltonian Mechanics, Springer, New York, 1978, 1982.

    Google Scholar 

  82. Schafer, R. D.: Introduction to Nonassociative Algebras, Academic Press, 1966.

  83. Schafer, R. D.: Generalized standard algebras, J. Algebra 12 (1969), 386-417.

    Google Scholar 

  84. Shaw, J. B.: Synopsis of linear associative algebras: a report on its natural development and results reached up to the present time, Carnegie Institute, Washington, 1907.

    Google Scholar 

  85. Slansky, R.: Group theory for unified model building, Phys. Rep. 79 (1981), 1-128.

    Google Scholar 

  86. Solodovnikova, E. P., Tavkhelidze, A. N. and Khrustalev, O. A.: Bogolyubov transformation in the strong coupling theory, II, Preprint P-6115, JINR, Dubna, 1971 (in Russian).

    Google Scholar 

  87. Sommerfeld, A.: Atombau und Spektrallinien, II, Friedr. Vieweg & Sohn, Braunschweig, 1951 (Ch. 4 and Math. Addendum 13).

    Google Scholar 

  88. Sorgsepp, L. and Lõhmus, J.: About nonassociativity in physics and Cayley-Graves' octonions, Hadronic J. 2 (1979), 1388-1459.

    Google Scholar 

  89. Sorgsepp, L. and Lõhmus, J.: Binary and ternary sedenions, Hadronic J. 4 (1981), 327-353.

    Google Scholar 

  90. Sorgsepp, L. and Lõhmus, J.: Dirac equation in the regular bimodule representation of octonions, in: P. Kuusk and J. Lõhmus (eds), Fundamental Interactions, Trans. Inst. Phys. Acad. Sci. Estonian SSR 62, Tartu, 1987, pp. 159-173.

  91. Sorgsepp, L. and Lõhmus, J.: Associator quantization and the deep structure of matter, in Trans. Tallinn Tech. Univ. No. 733, Tallinn, 1992, pp. 85-92.

  92. Sorgsepp, L. and Lõhmus, J.: Fundamental fermions in a nonassociative model of matter, in I. Ots and L. Palgi (eds), Proc. 2nd Tallinn Symposium on Neutrino Physics, held at Lohusalu, Tallinn, Estonia, Oct. 5-8, 1993, Estonian Acad. Sci., Inst. of Physics, Tartu, 1994, pp. 181-188.

    Google Scholar 

  93. Study, E. and Cartan, E.: Nombres complexes, in Encycl. Sci. Math. Pures Appl. I. Vol. 1. Arithmétique, Gauthier-Villars & Teubner, Paris-Liepzig, 1904, pp. 324-468.

    Google Scholar 

  94. Tamas, V.: Properties of a non-associative ternary structure, An. Sti. Univ. Iasi (Suppl.) 31 (1985), 51-53.

    Google Scholar 

  95. Tomber, M. L.: A short history of nonassociative algebras, Hadronic J. 2 (1979), 507-725.

    Google Scholar 

  96. Tomber, M. L., Norris, D. M., Reynolds, M., Balzer, C., Trebilcott, K., Terry, T., Coryell, H. and Ordway, J.: A nonassociative algebra bibliography, Hadronic J. 3 (1979), 507-725.

    Google Scholar 

  97. Tomber, M. L., Norris, D. M. and Smith, C. L.: A subject index of works relating to nonassociative algebras, Hadronic J. 4 (1981), 1444-1625.

    Google Scholar 

  98. Tomber, M. L., Smith, C. L., Norris, D. M. and Welk, R.: Addenda to 'A nonassociative algebra bibliography', Hadronic J. 4 (1981), 1328-1443.

    Google Scholar 

  99. Tomber, M. L.: The history and methods of Lie-admissible algebras, Hadronic J. 5 (1982), 360-430.

    Google Scholar 

  100. van der Waerden, B. L.: A History of Algebra from al-Khwarizmi to Emmy Noether, Springer, New York, 1985.

    Google Scholar 

  101. van Nieuwenhuisen, P.: Supergravity, Phys. Rep. 68 (1981), 89-398.

    Google Scholar 

  102. Wedderburn, J. H.: On hypercomplex numbers, Proc. London Math. Soc. (2) 6 (1907), 77-118.

    Google Scholar 

  103. West, P.: Introduction to Supersymmetry and Supergravity, World Scientific, Singapore, 1986.

    Google Scholar 

  104. Wörz-Busekros, A.: Algebras in Genetics, Lecture Notes in Biomath. 36, Sringer, New York, 1980.

    Google Scholar 

  105. Yamaguti, K.: On the theory of Malcev algebras, Kumamoto J. Sci. A 6 (1963), 9-45.

    Google Scholar 

  106. Zhevlakov, K. A., Slin'ko, A. M., Shestakov, I. P. and Shirshov, A. I.: Nearly Associative Rings, Nauka, Moscow, 1978 (in Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lõhmus, J., Paal, E. & Sorgsepp, L. About Nonassociativity in Mathematics and Physics. Acta Applicandae Mathematicae 50, 3–31 (1998). https://doi.org/10.1023/A:1005854831381

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1005854831381