Bulletin of the American Mathematical Society

Structure and representation of nonassociative algebras

R. D. Schafer

Full-text: Open access

Article information

Source
Bull. Amer. Math. Soc. Volume 61, Number 6 (1955), 469-484.

Dates
First available: 4 July 2007

Permanent link to this document
http://projecteuclid.org/euclid.bams/1183520113

Mathematical Reviews number (MathSciNet)
MR0072125

Zentralblatt MATH identifier
0066.02301

Citation

Schafer, R. D. Structure and representation of nonassociative algebras. Bulletin of the American Mathematical Society 61 (1955), no. 6, 469--484. http://projecteuclid.org/euclid.bams/1183520113.


Export citation

References

  • 1. A. A. Albert, On a certain algebra of quantum mechanics, Ann. of Math. vol. 35 (1934) pp. 65-73.
  • 2. A. A. Albert, Quadratic forms permitting composition, Ann. of Math. vol. 43 (1942) pp. 161-177.
  • 3. A. A. Albert, Non-associative algebras, I, Ann. of Math. vol. 43 (1942) pp. 685-707; II, ibid. pp. 708-723.
  • 4. A. A. Albert, The radical of a non-associative algebra, Bull. Amer. Math. Soc. vol. 48 (1942) pp. 891-897.
  • 5. A. A. Albert, On Jordan algebras of linear transformations, Trans. Amer. Math. Soc. vol. 59 (1946) pp. 524-555.
  • 6. A. A. Albert, A structure theory for Jordan algebras, Ann. of Math. vol. 48 (1947) pp. 546-567.
  • 7. A. A. Albert, On the power-associativity of rings, Summa Brasiliensis Mathematicae vol. 2 (1948) pp. 21-32.
  • 8. A. A. Albert, Power-associative rings, Trans. Amer. Math. Soc. vol. 64 (1948) pp. 552-593.
  • 9. A. A. Albert, On right alternative algebras, Ann. of Math. vol. 50 (1949) pp. 318-328.
  • 10. A. A. Albert, A theory of trace-admissible algebras, Proc. Nat. Acad. Sci. U.S.A. vol. 35 (1949) pp. 317-322.
  • 11. A. A. Albert, Absolute-valued algebraic algebras, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 763-768; A note of correction, Bull. Amer. Math. Soc. vol. 55 (1949) p. 1191[Note].
  • 12. A. A. Albert, A note on the exceptional Jordan algebra, Proc. Nat. Acad. Sci. U.S.A. vol. 36 (1950) pp. 372-374.
  • 13. A. A. Albert, A theory of power-associative commutative algebras, Trans. Amer. Math. Soc. vol. 69 (1950) pp. 503-527.
  • 14. A. A. Albert, New simple power-associative algebras, Summa Brasiliensis Mathematicae vol. 2 (1951) pp. 183-194.
  • 15. A. A. Albert, Power-associative algebras, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, Providence, American Mathematical Society, 1952, vol. 2, pp. 2-32.
  • 16. A. A. Albert, On simple alternative rings, Canadian Journal of Mathematics vol. 4 (1952) pp. 129-135.
  • 17. A. A. Albert, On nonassociative division algebras, Trans. Amer. Math. Soc. vol. 72 (1952) pp. 296-309.
  • 18. A. A. Albert, On commutative power-associative algebras of degree two, Trans. Amer. Math. Soc. vol. 74 (1953) pp. 323-343.
  • 19. A. A. Albert, The structure of right alternative algebras, Ann. of Math. (2) vol. 59 (1954) pp. 408-417.
  • 20. V. G. Askinuze, A theorem on the splittability of J-algebras, Ukrain Mat. Zurnal. vol. 3 (1951) pp. 381-398 (Russian).
  • 21. E. A. Behrens, Nichtassoziative Ringe, Math. Ann. vol. 127 (1954) pp. 441-452.
  • 22. G. Birkhoff and P. M. Whitman, Representations of Jordan and Lie algebras, Trans. Amer. Math. Soc. vol. 65 (1949) pp. 116-136.
  • 23. R. H. Bruck, Some results in the theory of linear non-associative algebras, Trans. Amer. Math. Soc. vol. 56 (1944) pp. 141-199.
  • 24. R. H. Bruck and E. Kleinfeld, The structure of alternative division rings, Proc. Amer. Math. Soc. vol. 2 (1951) pp. 878-890.
  • 25. H. E. Campbell, An extension of the "principal theorem" of Wedderburn, Proc. Amer. Math. Soc. vol. 2 (1951) pp. 581-585.
  • 26. A. Cayley, On Jacobi's elliptic functions, in reply to the Rev. Brice Bronwin; and on quaternions, Philosophical Magazine vol. 26 (1845) pp. 210-213.
  • 27. C. Chevalley and R. D. Schafer, The exceptional simple Lie algebras F4 and E6, Proc. Nat. Acad. Sci. U.S.A. vol. 36 (1950) pp. 137-141.
  • 28. P. M. Cohn, On homomorphic images of special Jordan algebras, Canadian Journal of Mathematics vol. 6 (1954) pp. 253-264.
  • 29. L. E. Dickson, On quaternions and their generalization and the history of the eight square theorem, Ann. of Math. vol. 20 (1919) pp. 155-171.
  • 30. R. Dubisch and S. Perlis, On the radical of a non-associative algebra, Amer. J. Math. vol. 70 (1948) pp. 540-546.
  • 31. S. Eilenberg, Extensions of general algebras, Annales de la Société Polonaise de Mathématique vol. 21 (1948) pp. 125-134.
  • 32. I. M. H. Etherington, Genetic algebras, Proceedings of the Royal Society of Edinburgh vol. 59 (1939) pp. 242-258.
  • 33. I. M. H. Etherington, Special train algebras, Quart. J. Math. Oxford Ser. vol. 12 (1941) pp. 1-8.
  • 34. I. M. H. Etherington, Non-associative algebra and the symbolism of genetics, Proceedings of the Royal Society of Edinburgh vol. 61 (1941) pp. 24-42.
  • 35. H. Freudenthal, Oktaven, Ausnahmengruppen und Oktavengeometrie, Utrecht, 1951.
  • 36. M. Hall, Projective planes, Trans. Amer. Math. Soc. vol. 54 (1943) pp. 229-277. Correction to Projective planes, ibid. vol. 65 (1949) p. 474.
  • 37. Harish-Chandra, On the radical of a Lie algebra, Proc. Amer. Math. Soc. vol. 1 (1950) pp. 14-17.
  • 38. G. P. Hochschild, Semi-simple algebras and generalized derivations, Amer. J. Math. vol. 64 (1942) pp. 677-694.
  • 39. H. Hopf, Ein topologischer Beitrag zur reellen Algebra, Comment. Math. Helv. vol. 13 (1941) pp. 219-239.
  • 40. F. D. Jacobson and N. Jacobson, Classification and representation of semi-simple Jordan algebras, Trans. Amer. Math. Soc. vol. 65 (1949) pp. 141-169.
  • 41. N. Jacobson, A note on non-associative algebras, Duke Math. J. vol. 3 (1937) pp. 544-548.
  • 42. N. Jacobson, Abstract derivation and Lie algebras, Trans. Amer. Math. Soc. vol. 42 (1937) pp. 206-224.
  • 43. N. Jacobson, Cayley numbers and normal simple Lie algebras of type G, Duke Math. J. vol. 5 (1939) pp. 775-783.
  • 44. N. Jacobson, Structure theory of simple rings without finiteness assumptions, Trans. Amer. Math. Soc. vol. 57 (1945) pp. 228-245.
  • 45. N. Jacobson, Isomorphisms of Jordan rings, Amer. J. Math. vol. 70 (1948) pp. 317-326.
  • 46. N. Jacobson, The center of a Jordan ring, Bull. Amer. Math. Soc. vol. 54 (1948) pp. 316-322.
  • 47. N. Jacobson, Derivation algebras and multiplication algebras of semi-simple Jordan algebras, Ann. of Math. vol. 50 (1949) pp. 866-874.
  • 48. N. Jacobson, Lie and Jordan triple systems, Amer. J. Math. vol. 71 (1949) pp. 149-170.
  • 49. N. Jacobson, General representation theory of Jordan algebras, Trans. Amer. Math. Soc. vol. 70 (1951) pp. 509-530.
  • 50. N. Jacobson, Representation theory for Jordan rings, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, Providence, American Mathematical Society, 1952, vol. 2, pp. 37-43.
  • 51. N. Jacobson, Une generalisation du théorème d'Engel, C. R. Acad. Sci. Paris vol. 234 (1952) pp. 579-581.
  • 52. N. Jacobson, Operator commutativity in Jordan algebras, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 973-976.
  • 53. N. Jacobson, Structure of alternative and Jordan bimodules, Osaka Math. J. vol. 6 (1954) pp. 1-71.
  • 54. N. Jacobson, A Kronecker factorization theorem for Cayley algebras and the exceptional simple Jordan algebra, Amer. J. Math. vol. 76 (1954) pp. 447-452.
  • 55. N. Jacobson and C. E. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. vol. 69 (1950) pp. 479-502.
  • 56. N. Jacobson, Homomorphisms of Jordan rings of self-adjoint elements, Trans. Amer. Math. Soc. vol. 72 (1952) pp. 310-322.
  • 57. W. E. Jenner, The radical of a non-associative ring, Proc. Amer. Math. Soc. vol. 1 (1950) pp. 348-351.
  • 58. P. Jordan, Über eine Klasse nichtassoziativer hyperkomplexer Algebren, Nachr. Ges. Wiss. Göttingen, 1932, pp. 569-575.
  • 59. P. Jordan, Über Verallgemeinerungsmöglichkeiten des Formalismus der Quantenmechanik, Nachr. Ges. Wiss. Göttingen (1933) pp. 209-214.
  • 60. P. Jordan, Über die Multiplication quantenmechanischer Grössen, Zeitschrift für Physik vol. 80 (1933) pp. 285-291.
  • 61. P. Jordan, J. von Neumann, and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. of Math. vol. 35 (1934) pp. 29-64.
  • 62. G. K. Kalisch, On special Jordan algebras, Trans. Amer. Math. Soc. vol. 61 (1947) pp. 482-494.
  • 63. I. Kaplansky, Semisimple alternative rings, Portugaliae Mathematica vol. 10 (1951) pp. 37-50.
  • 64. I. Kaplansky, Infinite-dimensional quadratic forms permitting composition, Proc. Amer. Math. Soc. vol. 4 (1953) pp. 956-960.
  • 65. E. Kleinfeld, Alternative division rings of characteristic 2, Proc. Nat. Acad. Sci. U.S.A. vol. 37 (1951) pp. 818-820.
  • 66. E. Kleinfeld, Right alternative rings, Proc. Amer. Math. Soc. vol. 4 (1953) pp. 939-944.
  • 67. E. Kleinfeld, Simple alternative rings, Ann. of Math. vol. 58 (1953) pp. 544-547.
  • 68. L. A. Kokoris, Power-associative commutative algebras of degree two, Proc. Nat. Acad. Sci. U.S.A. vol. 38 (1952) pp. 534-537.
  • 69. L. A. Kokoris, New results on power-associative algebras, Trans. Amer. Math. Soc. vol. 77 (1954) pp. 363-373.
  • 70. A. Kurosch, Non-associative free algebras and free products of algebras, Rec. Math. (Mat. Sbornik) N.S. vol. 20 (62) (1947) pp. 239-262 (Russian).
  • 71. A. Kurosch, The present status of the theory of rings and algebras, Uspehi Matematičeskih Nauk (N.S.) vol. 6, no. 2 (42) (1951) pp. 3-15 (Russian).
  • 72. W. G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. vol. 72 (1952) pp. 217-242.
  • 73. A. Malcev, On the representation of an algebra as a direct sum of the radical and a semisimple subalgebra, C. R. (Doklady) Acad. Sci. URSS vol. 36 (1942) pp. 42-45.
  • 74. A. Malcev, On a representation of nonassociative rings, Uspehi Matematičeskih Nauk (N.S.) vol. 7, no. 1 (47) (1952) pp. 181-185 (Russian).
  • 75. R. Moufang, Alternativkörper und der Satz vom vollständigen Vierseit(D9), Abh. Math. Sem. Hamburgischen Univ. vol. 9 (1933) pp. 207-222.
  • 76. R. Moufang, Zur Struktur von Alternativkörpern, Math. Ann. vol. 110 (1935) pp. 416-430.
  • 77. W. H. Mills, A theorem on the representation theory of Jordan algebras, Pacific Journal of Mathematics vol. 1 (1951) pp. 255-264.
  • 78. B. H. Neumann, Embedding non-associative rings in division rings, Proc. London Math. Soc. (3) vol. 1 (1951) pp. 241-256.
  • 79. L. J. Paige, A theorem on power-associative loop algebras, Proc. Amer. Math. Soc. vol. 6 (1955) pp. 279-280.
  • 80. A. J. Penico, The Wedderburn principal theorem for Jordan algebras, Trans. Amer. Math. Soc. vol. 70 (1951) pp. 404-420.
  • 81. C. M. Price, Jordan division algebras and the algebras A(λ), Trans. Amer. Math. Soc. vol. 70 (1951) pp. 291-300.
  • 82. R. L. San Soucie, Right alternative division rings of characteristic 2, Proc. Amer. Math. Soc. vol. 6 (1955) pp. 291-296.
  • 83. R. D. Schafer, Alternative algebras over an arbitrary field, Bull. Amer. Math. Soc. vol. 49 (1943) pp. 549-555.
  • 84. R. D. Schafer, Concerning automorphisms of non-associative algebras, Bull. Amer. Math. Soc. vol. 53 (1947) pp. 573-583.
  • 85. R. D. Schafer, The exceptional simple Jordan algebras, Amer. J. Math. vol. 70 (1948) pp. 82-94.
  • 85. R. D. Schafer, Structure of genetic algebras, Amer. J. Math. vol. 71 (1949) pp. 121-135.
  • 87. R. D. Schafer, The Wedderburn principal theorem for alternative algebras, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 604-614.
  • 88. R. D. Schafer, Inner derivations of non-associative algebras, Bull. Amer. Math. Soc. vol. 55 (1949) pp. 769-776.
  • 89. R. D. Schafer, A theorem on the derivations of Jordan algebras, Proc. Amer. Math. Soc. vol. 2 (1951) pp. 290-294.
  • 90. R. D. Schafer, Representations of alternative algebras, Trans. Amer. Math. Soc. vol. 72 (1952) pp. 1-17.
  • 91. R. D. Schafer, The Casimir operation for alternative algebras, Proc. Amer. Math. Soc. vol. 73 (1953) pp. 444-451.
  • 92. R. D. Schafer, A generalization of a theorem of Albert, Proc. Amer. Math. Soc. vol. 73 (1953) pp. 452-455.
  • 93. R. D. Schafer, On the algebras formed by the Cayley- Dicks on process, Amer. J. Math. vol. 76 (1954) pp. 435-446.
  • 94. R. D. Schafer, Noncommutative Jordan algebras of characteristic 0, Proc. Amer. Math. Soc. vol. 6 (1955) pp. 472-475.
  • 95. L. A. Skornyakov, Natural domains of Veblen-Wedderburn projective planes, Izvestiya Akad. Nauk SSSR vol. 13 (1949) pp. 447-472 (Russian). Amer. Math. Soc. Translation no. 58.
  • 96. L. A. Skornyakov, Alternative fields, Ukrain. Mat. Zurnal vol. 2 (1950) pp. 70-85 (Russian).
  • 97. L. A. Skornyakov, Right-alternative fields, Izvestiya Akad. Nauk SSSR Ser. Mat. vol. 15 (1951) pp. 177-184 (Russian).
  • 98. L. A. Skornyakov, Projective planes, Uspehi Matematičeskih Nauk (N.S.) vol. 6 (1951) pp. 112-154 (Russian). Amer. Math. Soc. Translation no. 99.
  • 99. M. F. Smiley, The radical of an alternative ring, Ann. of Math. vol. 49 (1948) pp. 702-709.
  • 100. M. F. Smiley, Application of a radical of Brown and McCoy to non-associative rings, Amer. J. Math. vol. 72 (1950) pp. 93-100.
  • 101. M. F. Smiley, On the ideals and automorphisms of non-associative rings, Proc. Amer. Math. Soc. vol. 2 (1951) pp. 138-143.
  • 102. M. F. Smiley, Some questions concerning alternative rings, Bull. Amer. Math. Soc. vol. 57 (1951) pp. 36-43.
  • 103. E. Stiefel, On Cauchy-Riemann equations in higher dimensions, Journal of Research of the National Bureau of Standards vol. 48 (1952) pp. 395-398.
  • 104. O. Taussky, An algebraic property of Laplace's differential equation, Quart. J. Math. vol. 10 (1939) pp. 99-103.
  • 105. M. L. Tomber, Lie algebras of type F, Proc. Amer. Math. Soc. vol. 4 (1953) pp. 759-768.
  • 106. J. von Neumann, On an algebraic generalization of the quantum mechanical formalism, Mat. Sbornik vol. 1 (1936) pp. 415-482.
  • 107. F. B. Wright, Absolute valued algebras, Proc. Nat. Acad. Sci. U.S.A. vol. 39 (1953) pp. 330-332.
  • 108. J. L. Zemmer, Jr., On the subalgebras of finite division algebras, Canadian Journal of Mathematics vol. 4 (1952) pp. 491-503.
  • 109. M. Zorn, Theorie der alternativen Ringe, Abh. Math. Sem. Hamburgischen Univ. vol. 8 (1930) pp. 123-147.
  • 110. M. Zorn, Alternativkörper und quadratische systeme, Abh. Math. Sem. Hamburgischen Univ. vol. 9 (1933) pp. 395-402.
  • 111. M. Zorn, The automorphisms of Cayley's non-associative algebra, Proc. Nat. Acad. Sci. U.S.A. vol. 21 (1935) pp. 355-358.
  • 112. M. Zorn, Alternative rings and related questions, I: Existence of the radical, Ann. of Math. vol. 42 (1941) pp. 676-686.