Unitary representations of non-compact supergroups - Communications in Mathematical Physics
- ️Günaydin, M.
- ️Tue Mar 01 1983
References
For a review of the theory of unitary representations of non-compact groups see: Schmid, W.: Proc. of the International Congress of Mathematicians, Helsinki (1978); Academia Scientiarum Fennica (Helsinki, 1980)
Wess, J., Zumino, B.: Supergauge transformations in four dimensions. Nucl. Phys. B70, 39 (1974)
Volkov, D.V., Akulov, V.P.: Is the neutrino a Goldstone particle? Phys. Lett.46B, 109 (1973)
Kac, V.G.: Adv. Math.26, 8 (1977)
See also, Freund, P.G.O., Kaplansky, I.: Simple supersymmetries. J. Math. Phys.17, 228 (1976)
Kac, V.G.: Representations of classical Lie superalgebras. In: Differential geometrical methods in mathematical physics. Bleuler, K., Petry, H.R., Reetz, A. (eds.). Berlin, Heidelberg, New York: Springer 1978
Ne'eman, Y., Sternberg, S.: Internal supersymmetry and unification. Proc. Natl. Acad. Sci. (USA)77, 3127 (1980)
Marcu, M.: The representations of Spl(2,1) — an example of representations of basic superalgebras. J. Math. Phys.21, 1277 (1980)
Dondi, P.H., Jarvis, P.D.: Assignments in strong-electroweak unified models with internal and spacetime supersymmetry. Z. Physik C4, 201 (1980)
Balantekin, A.B., Bars, I.: Dimension and character formulas for Lie supergroups, J. Math. Phys.22, 1149 (1981), Representations of supergroups. J. Math. Phys.22, 1810 (1981) and Branching Rules for the Supergroup SU(N/M) from those of SU(N + M). J. Math. Phys.23, 1239 (1982)
For a comprehensive review and further references to the representations of compact supergroups see: Bars, I.: Supergroups and their Representations, Yale preprint YTP-8-25, Proceedings of School on Supersymmetry in Physics, Mexico (December 1981) (to be published)
Balantekin, A.B., Bars, I., Iachello, F.: Nucl. Phys. A370, 284 (1981)
Balantekin, A.B.: Ph. D. Thesis, Yale University (1982) (unpublished)
Bars, I., Morel, B., Ruegg, H.: CERN preprint TH-3333 (1982) to appear in J. Math. Phys. (1983)
Cremmer, E., Ferrara, S., Scherk, J.: SU(4) invariant supergravity theory. Phys. Lett.74, 61 (1978)
Cremmer, E., Julia, B.: The SO(8) supergravity. Nucl. Phys. B159, 141 (1979)
Günaydin, M., Saçlioğlu, C.: Bosonic construction of the Lie algebras of some non-compact groups appearing in the supergravity theories and their oscillator-like unitary representations. Phys. Lett.108B, 180 (1982)
Günaydin, M., Saçlioğlu, C.: Oscillator-like unitary representations of non-compact groups with a Jordan structure and the non-compact groups of supergravity. Commun. Math. Phys.87, 159–179 (1982)
For a discussion of the relevance of the unitary representations given in [13] and [14] to the attempts to extract a realistic grand unified theory from supergravity theories, see: Günaydin, M.: Unitary realizations of non-compact groups of supergravity, talk presented at the Second Europhysics Study Conference on Unification of Fundamental Interactions, Erice, Sicily (1981); CERN preprint TH-3222 (1981)
Loebl, E.M. (ed.): Group theory and its applications, Vols. I–III. New York: Academic Press 1968
Wybourne, B.G.: Classical groups for physicists. New York: Wiley 1974
Dyson, F.J.: Symmetry groups in nuclear and particle physics. New York: Benjamin 1966
Gürsey, F. (ed.): Group theoretical concepts and methods in elementary particle physics. New York: Gordon and Breach 1964
Bargmann, V.: Ann. Math.48, 568 (1947); Commun. Pure Appl. Math.14, 187 (1961)
Kashiwara, M., Vergne, M.: Invent. Math.44, 1 (1978)
Howe, R.: Classical invariant theory. Yale Univ. preprint, unpublished; and Transcending classical invariant theory. Yale Univ. preprint (unpublished)
R. Howe has suggested the possibility of extending the dual pair notion to the case of superalgebras (private communication). See also [19]
We should note that when the generators are represented by ordinary matrices rather than oscillators,θ α will, of course, commute with those matrices
Gürsey, F., Marchildon, L.: Spontaneous symmetry breaking and nonlinear invariant Lagrangians: Applications to SU(2)⊗U(2) and OSp(1/4). Phys. Rev. D17, 2038 (1978); The graded Lie groups SU(2,2/1) and OSp(1/4). J. Math. Phys.19, 942 (1978)
For a review of coherent states and their applications, see: Perelomov, A.M.: Sov. Phys. Usp.20, 703 (1977)
For a study of the analyticity properties of coherent state representations of Lie groups and further references on the subject, see: Onofri, E.: A note on coherent state representations of Lie groups. J. Math. Phys.16, 1087 (1974)
Gelfand, I.M., Graev, M.I., Vilenkin, N.Y.: Generalized functions, Vol. 5. New York: Academic Press 1968. For an operator treatment of the unitary representations of some non-compact groups related to the Poincaré group à la Gelfand et al., see: Gürsey, F.: Representations of some non-compact groups related to the Poincaré group. Yale Univ. mimeographed notes (1971)
Bars, I., Gürsey, F.: Operator treatment of the Gelfand-Naimark basis for SL(2,C)*. J. Math. Phys.13, 131 (1972)
Gürsey, F., Orfanidis, S.: Conformal invariance and field theory in two dimensions. Phys. Rev. D7, 2414 (1973)
Tits, J.: Nederl. Akad. van Wetenschapp65, 530 (1962)
Koecher, M.: Am. J. Math.89, 787 (1967)
Bars, I., Günaydin, M.: Construction of Lie algebras and Lie superalgebras from ternary algebras. J. Math. Phys.20, 1977 (1979)
Bars, I.: Proceedings of the 8th Intern. Coll. on Group Theoretical Methods. Annals of Israel Physical Society, Vol. 3 (1980)
Günaydin, M.: Proceedings of the 8th Intern. Coll. on Group Theoretical Methods. Annals of Israel Physical Society, Vol. 3 (1980), p. 279
Kac, V.: Commun. Algebra5, 1375 (1977)