Pacific Journal of Mathematics

Superalgebras.

Irving Kaplansky

Source: Pacific J. Math. Volume 86, Number 1 (1980), 93-98.

Primary Subjects: 17A70
Secondary Subjects: 17A25, 81C40, 81G20

Full-text: Open access

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.pjm/1102780617
Zentralblatt MATH identifier: 0438.17003
Mathematical Reviews number (MathSciNet): MR586871

References

[1] B. L. Aneva, S. G. Mihov, and D. C. Stojanov, On some properties of representa- tions of conformal superalgebra, Theor. Mat. Fiz.y 31 (1977), 179-189 (Russian with
Mathematical Reviews (MathSciNet): MR58:32492
Zentralblatt MATH: 0411.17005
[2] R. Arnowitt and P. Nath, Spontaneous symmetry breaking of gauge super symmetry, Phys. Rev. Letters, 36 (1976), 1526-1529.
Mathematical Reviews (MathSciNet): MR54:6710
[3] Nigel Backhouse, Some aspects of graded Lie algebras, pp. 249-254 in Group Theoreti- cal Methods in Physics, Academic Press, New York, 1977.
Mathematical Reviews (MathSciNet): MR58:22209
[4] Nigel Backhouse, The Killing form for graded Lie algebras, J. Math. Physics, 18 (1977), 239-244.
Mathematical Reviews (MathSciNet): MR56:3073
Zentralblatt MATH: 0358.17009
[5] Nigel Backhouse, A classification of four-dimensionalLie super algebras, J. Math. Physics, 19 (1978), 2400-2402.
Mathematical Reviews (MathSciNet): MR80c:17004
Zentralblatt MATH: 0421.17002
[6] F. A. Berezin, Representations of the supergroup U(p,q), Functional Anal. Appl., 10 (1976), no. 3, 70-71; translation 221-223.
Mathematical Reviews (MathSciNet): MR54:10495
Zentralblatt MATH: 0342.22013
[7] F. A. Berezin and D. A. Letes, Super manifolds,Doklady Akad. Nauk SSSR, 224 (1975), 505-508; translation 16 (1975), 1218-1222.
Mathematical Reviews (MathSciNet): MR53:6609
Zentralblatt MATH: 0331.58005
[8] F. A. Berezin and V. S. Retah, The structure of Lie superalgebras with semisimple even part, Functional Anal. Appl., 12 (1978), no. 1, 64-65; translation 48-49.
Mathematical Reviews (MathSciNet): MR58:5815
Zentralblatt MATH: 0411.17003
[9] L. Corwin, Y. Ne'eman, and S. Sternberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Reviews of Modern Physics, 47 (1975), 573-603.
Mathematical Reviews (MathSciNet): MR55:11828
Zentralblatt MATH: 0557.17004
[10] Geoffrey Dixon, Fermi-Bose and internalsymmetrieswith universalClifford algebras, J. Math. Physics, 18 (1977), 2204-2206.
Mathematical Reviews (MathSciNet): MR57:8532
[11] D. Z. Djokovic, Classification of some 2-graded Lie algebras, J. Pure Applied Alg., 7 (1976), 217-230.
Mathematical Reviews (MathSciNet): MR52:10822
Zentralblatt MATH: 0319.17005
[12] D. Z. Djokovic, Representation theory for symplectic 2-gradedLie algebras, J. Pure Appl. Algebra, 9 (1976-7), 25-38.
Mathematical Reviews (MathSciNet): MR54:10353
Zentralblatt MATH: 0344.17005
[13] D. Z. Djokovic, Isomorphism of some simple 2-graded Lie algebras, Canad. J. Math., 29 (1977), 289-294.
Mathematical Reviews (MathSciNet): MR55:429
Zentralblatt MATH: 0343.17005
[14] D. Z. Djokovic and G. Hochschild, Semisimplicityof 2-graded Lie algebras II, Illinois J. Math., 20 (1976), 134-143.
Mathematical Reviews (MathSciNet): MR52:8206
Zentralblatt MATH: 0335.17016
[15] Mohamed El-Agawany and Artibano Micali, Le theoreme dePoincare-Birkhoff-Witt pour les algebres de Lie graduees, R. C. Acad. Sci. Paris, 285A (1977), 165-168.
Mathematical Reviews (MathSciNet): MR56:15714
Zentralblatt MATH: 0362.17009
[16] Peter G. O. Freund and I. Kaplansky, Simple super symmetries,J. Mathematical Phys., 17 (1976), 228-231.
Mathematical Reviews (MathSciNet): MR53:7249
Zentralblatt MATH: 0399.17005
[17] C. Fronsdal, Differentialgeometry in Grassman algebras, Letters in Math. Physics, 1 (1976), 165-170.
Mathematical Reviews (MathSciNet): MR54:11229
Zentralblatt MATH: 0336.53024
[18] F. Grsey and L. Marchildon, The graded Lie groups S7(2,2/1) and OSpd/4), J. Mathematical Phys., 19 (1978), 942-951.
Mathematical Reviews (MathSciNet): MR58:4015
Zentralblatt MATH: 0389.22020
[19] J. Hietarinta, Super symmetrygenerators of arbitrary spin, Physical Reviews D, 13 (1976), 838-850.
[20] G. Hochschild, Semisimplicityof2-graded Lie algebras, Illinois J. Math., 2O (1976), 107-123.
Mathematical Reviews (MathSciNet): MR52:8205
Zentralblatt MATH: 0356.17009
[21] N. Jacobson, Lie Algebras, Interscience, 1962.
Mathematical Reviews (MathSciNet): MR26:1345
[22] V. G. Kac, Classification of simple Lie superalgebras, Functional Anal. Appl., 9 (1975), 3, 91-92; translation 263-265.
Mathematical Reviews (MathSciNet): MR52:10833
Zentralblatt MATH: 0351.17014
[23] V. G. Kac, Letter to the editor, Functional Anal. Appl., 10 (1976), no. 2 93; translation 163.
Zentralblatt MATH: 1016.30001
[24] V. G. Kac, A sketch of Lie superalgebra theory, Comm. Math. Phys., 53 (1977), 31-64.
Mathematical Reviews (MathSciNet): MR56:437
Zentralblatt MATH: 0359.17009
[25] V. G. Kac, Lie superalgebras, Advances in Math., 26 (1977), no. 1, 8-96.
Mathematical Reviews (MathSciNet): MR58:5803
[26] V. G. Kac, Characters of typicalrepresentationsof classical Liesuperalgebras, Comm. in Algebra, 5 (1977), 889-897.
Mathematical Reviews (MathSciNet): MR56:3075
Zentralblatt MATH: 0359.17010
[27] V. G. Kac,Classification of simple Z-graded Lie superalgebras and simpleJordan superalgebras, Comm. in Algebra, 5 (1977), 1375-1400.
Mathematical Reviews (MathSciNet): MR58:16806
Zentralblatt MATH: 0367.17007
[28] I. Kaplansky, Lie Algebras and Locally Compact Groups, Chicago Lectures in Mathematics, Univ. of Chicago Press, 1971.
Zentralblatt MATH: 0223.17001
[29] I. Kaplansky, Lie and Jordan superalgebras, to appear in Proc. of Charlottesville Conf.
[30] B. G. Konopel'cenko, Extensions of the Poincare algebra by spinorgenerators, JETP Letters, 21 (1975),612-614; translation 287-288.
[31] B. Kostant, Graded manifolds, graded Lie theory, and pr^quantization, Diff. Geom. Meth. Math. Phys., Proc. Symp. Bonn 1975, Springer Lecture Notes 570 (1977), 177-306.
Mathematical Reviews (MathSciNet): MR58:28326
Zentralblatt MATH: 0358.53024
[32] D. A. Leites, Cohomology of Lie super algebras, Functional Anal, and its AppL, 9 (1975), no. 4, 75-76; translation 340-341.
Mathematical Reviews (MathSciNet): MR54:10362
Zentralblatt MATH: 0586.17005
[33] D. J. R. Lloyd-Evans, Geometric aspects of supergauge theory, J. of Math. Physics 18 (1977), 1923-1927.
Mathematical Reviews (MathSciNet): MR57:8908
[34] F. Mansouri, A new class of superalgebras and local gauge groups in super space, J. of Math. Physics, 18 (1977), 2395-2396.
Mathematical Reviews (MathSciNet): MR58:28104
Zentralblatt MATH: 0383.17004
[35] J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, Bull. Amer. Math. Soc, 71 (1965), 372-377.
Mathematical Reviews (MathSciNet): MR32:3059
Zentralblatt MATH: 0163.03102
[36] J. P. May, Same title, J. of Alg., 3 (1966), 123-146.
[37] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math., 81 (1965), 211-264.
Mathematical Reviews (MathSciNet): MR30:4259
Zentralblatt MATH: 0163.28202
[38] W. Nahm, V. Rittenberg, and M. Scheunert, The classification of graded Lie algebras, Phys. Letters, 61B (1976), 383-384.
Mathematical Reviews (MathSciNet): MR53:15144
Zentralblatt MATH: 0335.17015
[39] W. Nahm, Graded Lie algebras: Generalization of Hermitianrepresentations,J. of Math. Physics, 18 (1977), 146-154.
Mathematical Reviews (MathSciNet): MR54:12844
Zentralblatt MATH: 0354.17004
[40] W. Nahm, Irreducible representationsof the osp(2,1) and spl(2t 1) graded Lie algbras, ibid., 155-162.
Mathematical Reviews (MathSciNet): MR54:12845
[41] A. Pais and V. Rittenberg, Semi-simple graded Lie algebras, J. Math. Physics, 16 (1975), 2062-2073; erratum ibid., 17 (1976), 598.
Mathematical Reviews (MathSciNet): MR53:13330b
[42] N. T. Petrov and R. P. Zaikov, Superalgebras, C. R. Acad. Bulgare Sci., 29 (1976), 1241-1243 (reviewed in Zbl. 366, no. 17014).
Mathematical Reviews (MathSciNet): MR56:1990
[43] V. S. Retah, Massey operations in Lie superalgebras and deformationsof complex- analytic algebras, Functional Anal. AppL, 11 (1977), no. 4, 88-89; translation 319-321.
Mathematical Reviews (MathSciNet): MR58:22670
[44] L. E. Ross, Representations of graded Lie algebras, Trans. Amer. Math. Soc, 120 (1965), 17-23.
Mathematical Reviews (MathSciNet): MR32:2513
Zentralblatt MATH: 0145.25903
[45] M. Scheunert, W. Nahm, and V. Rittenberg, Classification of all simple graded Lie algebras whose Lie algebra is reductive.I, J. Math. Physics, 17 (1976), 1626-1639, II, ibid., 1640-1644.
[46] G. Seligman, On Lie algebras of prime characteristic, Memoirs Amer. Math. Soc, no. 19, 1956.
Mathematical Reviews (MathSciNet): MR17:1108c
Zentralblatt MATH: 0071.02703
[47] S. Sternberg, Some recent results on super symmetry,Diff. Geom. Math. Phys., Symp. Bonn 1975, Springer Lecture Notes, 570 (1977), 145-176.
Mathematical Reviews (MathSciNet): MR56:8756
Zentralblatt MATH: 0351.17003
[48] S. Sternberg and J. A. Wolf, HermitianLie algebras and metaplectic representa- tions, Trans. Amer. Math. Soc, 238 (1978), 1-43.
Mathematical Reviews (MathSciNet): MR58:6081
Zentralblatt MATH: 0386.22010
[49] H. Tilgner, Graded generalizations of Weyl and Clifford algebras, J. Pure Appl. Algebra, 1O (1977), 163-168.
Mathematical Reviews (MathSciNet): MR56:15720
[50] H. Tilgner, A graded generalization of Lie triples, J. Algebra, 47 (1977), 190-196.
Mathematical Reviews (MathSciNet): MR56:15719
Zentralblatt MATH: 0354.17003
[51] H. Tilgner, Extensions of Lie-graded algebras, J. Math. Physics, 18 (1977), 1987-1991.
Mathematical Reviews (MathSciNet): MR56:17539
Zentralblatt MATH: 0383.17014

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