×

Boundary conformal field theories, limit sets of Kleinian groups and holography. (English) Zbl 0986.81095

Summary: In this paper, based on the available mathematical works on geometry and topology of hyperbolic manifolds and discrete groups, some results of D. Z. Freedman et al. [Nucl. Phys. B 546, 96-118 (1999; Zbl 0944.81041)] are reproduced and broadly generalized. Among many new results, the possibility of extension of work of Belavin, Polyakov and Zamolodchikov to higher dimensions is investigated. Objections known in the physical literature against such an extension are removed and the possibility of an extension is convencingly demonstrated.

MSC:

81T50 Anomalies in quantum field theory
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
57M60 Group actions on manifolds and cell complexes in low dimensions

Citations:

Zbl 0944.81041
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Popov, A., Holomorphic Chern-Simons-Witten theory: from 2D to 4D conformal field theories, Nuclear Phys. B, 550, 585-621 (1999) · Zbl 0947.81105
[2] Fradkin, E.; Palchik, M., New developments in \(d\)-dimensional conformal quantum field theory, Phys. Rep., 300, 1-111 (1998)
[3] Witten, E., Quantum field theory and Jones polynomial, Commun. Math. Phys., 121, 351-399 (1989) · Zbl 0667.57005
[4] E. Witten, The central charge in 3 dimensions, in: Physics and Mathematics of Strings, World Scientific, Singapore, 1990.; E. Witten, The central charge in 3 dimensions, in: Physics and Mathematics of Strings, World Scientific, Singapore, 1990. · Zbl 0767.17023
[5] More, G.; Seiberg, N., Classical and quantum conformal field theory, Commun. Math. Phys., 123, 177-254 (1989) · Zbl 0694.53074
[6] Funar, L., (2+1)D topological quantum field theory and 2D conformal field theory, Commun. Math. Phys., 171, 405-458 (1995) · Zbl 0838.58047
[7] Bing, R.; Klee, V., Every simple curve in \(S^3\) is unknotted in \(S^4\), J. London Math. Soc., 39, 86-94 (1964) · Zbl 0116.40703
[8] B. Maskit, Kleinian Groups, Springer, Berlin, 1987.; B. Maskit, Kleinian Groups, Springer, Berlin, 1987.
[9] P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, Springer, Berlin, 1997.; P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, Springer, Berlin, 1997.
[10] Beardon, A.; Maskit, B., Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math., 132, 1-12 (1974) · Zbl 0277.30017
[11] Maldacena, J., The large \(N\) limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys., 2, 231-252 (1998) · Zbl 0914.53047
[12] Friedman, D. Z.; Mathur, S.; Matusis, A.; Rastelli, L., Correlation functions in the \(CFT_d/AdS_{d+1}\) correspondence, Nuclear Phys. B, 546, 96-118 (1999) · Zbl 0944.81041
[13] Witten, E., Anti de Sitter space and holography, Adv. Theor. Math. Phys., 2, 253-291 (1998) · Zbl 0914.53048
[14] Aref’eva, I.; Volovich, I., Breaking of conformal symmetry in the AdS/CFT correspondence, Phys. Lett. B, 433, 49-55 (1998)
[15] Mück, W.; Viswanathan, K., Conformal field theory correlators from classical scalar field theory on anti de Sitter space, Phys. Rev. D, 58, 41901 (1998)
[16] Mück, W.; Viswanathan, K., Conformal field theory correlators from classical field theory on anti de Sitter space: vector and spinor fields, Phys. Rev. D, 58, 106006 (1998)
[17] Susskind, L., The World as a hologram, J. Math. Phys., 36, 6377-6396 (1995) · Zbl 0850.00013
[18] G. t́ Hooft, Dimensional reduction in quantum gravity (qr-qc/9310026).; G. t́ Hooft, Dimensional reduction in quantum gravity (qr-qc/9310026).
[19] L. Susskind, E. Witten, The holographic bound in anti de Sitter space (hep-th/9805114).; L. Susskind, E. Witten, The holographic bound in anti de Sitter space (hep-th/9805114).
[20] C. Fefferman, C. Graham, Conformal invariants, Asterisque (hors serie) (1985) 92-116.; C. Fefferman, C. Graham, Conformal invariants, Asterisque (hors serie) (1985) 92-116.
[21] Graham, C.; Lee, J., Adv. Math., 87, 186-214 (1991)
[22] A. Petrov, Einsein Spaces, Pergamon Press, London, 1969.; A. Petrov, Einsein Spaces, Pergamon Press, London, 1969.
[23] T. Willmore, Riemannian Geometry, Clarendon Press, Oxford, 1993.; T. Willmore, Riemannian Geometry, Clarendon Press, Oxford, 1993.
[24] R. Kulkarni, Conformal structures and Möbius structures, in: Conformal Geometry, Vieweg, Wiesbaden, 1988, pp. 1-39.; R. Kulkarni, Conformal structures and Möbius structures, in: Conformal Geometry, Vieweg, Wiesbaden, 1988, pp. 1-39.
[25] L. Ahlfors, Möbius transformations in several dimensions, Lecture Notes, University of Minnesota, 1981.; L. Ahlfors, Möbius transformations in several dimensions, Lecture Notes, University of Minnesota, 1981.
[26] S. Hawking, G. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, 1973.; S. Hawking, G. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, 1973. · Zbl 0265.53054
[27] C. McMullen, Renormalization and 3-Manifolds which Fiber Over the Circle, Princeton University Press, Princeton, NJ, 1996.; C. McMullen, Renormalization and 3-Manifolds which Fiber Over the Circle, Princeton University Press, Princeton, NJ, 1996. · Zbl 0860.58002
[28] J. Elstrodt, F. Grunevald, J. Mennicke, Groups Acting on Hyperbolic Space, Springer, Berlin, 1998.; J. Elstrodt, F. Grunevald, J. Mennicke, Groups Acting on Hyperbolic Space, Springer, Berlin, 1998.
[29] J. Dozdziuk, J. Jorgenson, Spectral asymtotics on degenerating hyperbolic 3-manifolds, Mem. Amer. Math. Soc. 643 (1998).; J. Dozdziuk, J. Jorgenson, Spectral asymtotics on degenerating hyperbolic 3-manifolds, Mem. Amer. Math. Soc. 643 (1998).
[30] Sullivan, D., The Dirichlet problem at infinity for a negatively curved manifolds, J. Differential Geom., 18, 723-732 (1983) · Zbl 0541.53037
[31] Lyons, T.; Sullivan, D., Function theory, random paths and covering spaces, J. Differential Geom., 19, 299-323 (1984) · Zbl 0554.58022
[32] K. Matsutaki, M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford University Press, Oxford, 1998.; K. Matsutaki, M. Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Oxford University Press, Oxford, 1998.
[33] P. Nicholls, The Ergodic Theory of Discrete Groups, Cambridge University Press, Cambridge, 1989.; P. Nicholls, The Ergodic Theory of Discrete Groups, Cambridge University Press, Cambridge, 1989. · Zbl 0674.58001
[34] J. Morrow, K. Kodaira, Complex Manifolds, Holt, Rinehart & Winston, New York, 1971.; J. Morrow, K. Kodaira, Complex Manifolds, Holt, Rinehart & Winston, New York, 1971.
[35] A. Presley, G. Segal, Loop Groups, Oxford University Press, Oxford, 1986.; A. Presley, G. Segal, Loop Groups, Oxford University Press, Oxford, 1986.
[36] Belavin, A.; Polyakov, A.; Zamolodchikov, A., Infinite conformal symmetry in two dimensional quantum field theory, Nuclear Phys. B, 241, 333-380 (1984) · Zbl 0661.17013
[37] Kholodenko, A., Use of meanders and train tracks for description of defects and textures in liquid crystals and 2+1 gravity, J. Geom. Phys., 33, 23-58 (2000) · Zbl 0961.83016
[38] Kholodenko, A., Use of quadratic differentials for description of defects and textures in liquid crystals and 2+1 gravity, J. Geom. Phys., 33, 59-102 (2000) · Zbl 0961.83017
[39] Douglas, J., Green’s function and the problem of Plateau, Amer. J. Math., 61, 545-589 (1939) · JFM 65.0454.02
[40] S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, Berlin, 1992.; S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, Berlin, 1992. · Zbl 0765.31001
[41] Patterson, S., The limit set of a Fuchsian group, Acta Math., 136, 241-273 (1976) · Zbl 0336.30005
[42] Sullivan, D., The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ., 50, 171-202 (1979) · Zbl 0439.30034
[43] W. Thurston, Geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1979 (http://www.msri.org/gt3m/).; W. Thurston, Geometry and topology of 3-manifolds, Princeton University Lecture Notes, 1979 (http://www.msri.org/gt3m/).
[44] Bishop, C.; Jones, P., Hausdorff dimension and Kleinian groups, Acta Math., 179, 1-39 (1997) · Zbl 0921.30032
[45] Y. Imayoshi, M. Taniguchi, An Introduction to Teichmüller Spaces, Springer, Berlin, 1992.; Y. Imayoshi, M. Taniguchi, An Introduction to Teichmüller Spaces, Springer, Berlin, 1992. · Zbl 0754.30001
[46] Bers, L., Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc., 4, 257-300 (1972) · Zbl 0257.32012
[47] Canary, R.; Taylor, E., Kleinian groups with small limit set, Duke Math. J., 73, 371-381 (1994) · Zbl 0798.30030
[48] O. Mokhov, Symplectic and Poissonian structures on spaces of loops of smooth manifolds and integrable systems, Russian Math. Surveys 53 (1998) 86-192 (in Russian).; O. Mokhov, Symplectic and Poissonian structures on spaces of loops of smooth manifolds and integrable systems, Russian Math. Surveys 53 (1998) 86-192 (in Russian).
[49] Nag, S.; Verjovsky, A., Diff \((S^1)\) and the Teichmüller spaces, Commun. Math. Phys., 130, 123-138 (1990) · Zbl 0705.32013
[50] Ahlfors, L., Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of Math., 74, 171-191 (1961) · Zbl 0146.30602
[51] E. Cartan, Les espaces \(a\); E. Cartan, Les espaces \(a\)
[52] I. Gelfand, P. Milnos, Z. Shapiro, Representations of the Rotation and Lorentz Groups, Nauka, Moscow, 1958 (in Russian).; I. Gelfand, P. Milnos, Z. Shapiro, Representations of the Rotation and Lorentz Groups, Nauka, Moscow, 1958 (in Russian).
[53] Thomas, L., On unitary representations of the group of de Sitter space, Ann. of Math., 42, 113-126 (1941) · JFM 67.0079.02
[54] Newton, T., A note on the representation of the de Sitter group, Ann. of Math., 51, 730-733 (1950) · Zbl 0038.01702
[55] Takahashi, R., Sur les representations unitaires des groupes de Lorentz generalizes, Bull. Soc. Math. France, 91, 289-433 (1963) · Zbl 0196.15501
[56] Sudarshan, E.; Mukuda, N.; O’Raifeartaigh, L., Group theory of the Kepler problem, Phys. Lett., 19, 322-326 (1965)
[57] Bacry, H., The de Sitter group \(L_{4,1}\) and the bound states of the hydrogen atom, Il Nuovo Cim., 16, 322-334 (1966) · Zbl 0138.44701
[58] V. Kac, A. Raina, Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific, Singapore, 1987.; V. Kac, A. Raina, Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific, Singapore, 1987. · Zbl 0668.17012
[59] Bakalov, B.; Kac, V.; Voronov, A., Cohomology of conformal algebras, Commun. Math. Phys., 200, 561-598 (1999) · Zbl 0959.17018
[60] I. Gelfand, D. Fuks, Cohomologies of the Lie algebra of vector fields on the circle, Functional Anal. Appl. 2 (1968) 92-93 (in Russian).; I. Gelfand, D. Fuks, Cohomologies of the Lie algebra of vector fields on the circle, Functional Anal. Appl. 2 (1968) 92-93 (in Russian). · Zbl 0176.11501
[61] I. Aref́eva, I. Volovich, On large \(N\); I. Aref́eva, I. Volovich, On large \(N\)
[62] K. Scannell, Flat conformal structures and causality in de Sitter manifolds, Ph.D. Thesis, University of California, Los Angeles, CA, 1996.; K. Scannell, Flat conformal structures and causality in de Sitter manifolds, Ph.D. Thesis, University of California, Los Angeles, CA, 1996.
[63] Kholodenko, A., J. Math. Phys., 37, 1287-1313 (1996)
[64] C. Itzykson, J.-M. Drouffe, Statistical Field Theory, Vol. 1, Cambridge University Press, Cambridge, 1989.; C. Itzykson, J.-M. Drouffe, Statistical Field Theory, Vol. 1, Cambridge University Press, Cambridge, 1989. · Zbl 0825.81001
[65] A. Beardon, The Geometry of Discrete Groups, Springer, Berlin, 1983.; A. Beardon, The Geometry of Discrete Groups, Springer, Berlin, 1983. · Zbl 0528.30001
[66] P. Nicholls, in: T. Bedford, M. Keane, C. Series (Eds.), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, Oxford, 1992.; P. Nicholls, in: T. Bedford, M. Keane, C. Series (Eds.), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, Oxford, 1992.
[67] B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1982.; B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1982. · Zbl 0504.28001
[68] Sullivan, D., Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math., 153, 259-277 (1984) · Zbl 0566.58022
[69] Tukia, P., The Hausdorff dimension of the limit set of a geometrically finite Kleinian group, Acta Math., 152, 127-140 (1984) · Zbl 0539.30034
[70] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984.; S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984. · Zbl 0543.58001
[71] Beardon, A., The exponent of convergence of Poincaré series, Proc. London Math. Soc., 18, 461-483 (1968) · Zbl 0162.38801
[72] Kholodenko, A.; Vilgis, T., Some geometrical and topological problems in polymer physics, Phys. Rep., 298, 251-372 (1998)
[73] Bowditch, B., Geometrical finiteness for hyperbolic groups, J. Funct. Anal., 113, 245-317 (1993) · Zbl 0789.57007
[74] Patterson, S., The Laplacian operator on Riemann surface, Compositio Math., 31, 83-107 (1975) · Zbl 0321.30020
[75] Patterson, S., The Laplacian operator on Riemann surface, Compositio Math., 32, 71-112 (1976) · Zbl 0321.30021
[76] Patterson, S., The Laplacian operator on Riemann surface, Compositio Math., 33, 227-259 (1976) · Zbl 0342.30011
[77] Sullivan, D., Related aspects of positivity in Riemannian geometry, J. Differential Geom., 25, 327-351 (1987) · Zbl 0615.53029
[78] Patterson, S., Further remarks on the exponent of convergence of Poincaré series, Tohoku Math. J., 35, 357-373 (1983) · Zbl 0505.20036
[79] Davies, E.; Simon, B.; Taylor, M., \(L^p\) spectral theory of Kleinian groups, J. Funct. Anal., 78, 116-136 (1988) · Zbl 0644.58022
[80] Lax, P.; Phillips, R., Translation representation for automorphic solutions of the wave equation in non-Euclidean spaces I, Commun. Pure Appl. Math., 37, 303-328 (1984) · Zbl 0544.10024
[81] C. Epstein, The spectral theory of geometrically periodic hyperbolic 3-manifolds, Mem. Amer. Math. Soc. 335 (1985).; C. Epstein, The spectral theory of geometrically periodic hyperbolic 3-manifolds, Mem. Amer. Math. Soc. 335 (1985).
[82] Burger, M.; Canary, R., A lower bound on \(λ_0\) for geometrically finite hyperbolic \(n\)-manifolds, J. Reine Angew. Math., 454, 37-57 (1994) · Zbl 0806.53046
[83] Canary, R.; Minsky, Y.; Taylor, E., Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds, J. Geom. Anal., 9, 283-297 (1999)
[84] Grigoryan, A.; Noguchi, M., The heat kernel on hyperbolic space, Bull. London Math. Soc., 30, 643-650 (1998)
[85] W. Massey, Algebraic Topology: An Introduction, Springer, Berlin, 1967.; W. Massey, Algebraic Topology: An Introduction, Springer, Berlin, 1967. · Zbl 0153.24901
[86] R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry, Springer, Berlin, 1992.; R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry, Springer, Berlin, 1992. · Zbl 0768.51018
[87] H. Farkas, I. Kra, Riemann Surfaces, Springer, Berlin, 1992.; H. Farkas, I. Kra, Riemann Surfaces, Springer, Berlin, 1992. · Zbl 0764.30001
[88] Ahlfors, L., Finitely generated Kleinian groups, Amer. J. Math., 86, 413-429 (1964) · Zbl 0133.04201
[89] Sullivan, D., A finiteness theorem for cusps, Acta Math., 147, 289-299 (1981) · Zbl 0502.57004
[90] Abikoff, W., The Euler characteristic and inequalities for Kleinian groups, Proc. Amer. Math. Soc., 97, 593-601 (1986) · Zbl 0606.30045
[91] S. Katok, Fuchsian Groups, University of Chicago Press, Chicago, 1992.; S. Katok, Fuchsian Groups, University of Chicago Press, Chicago, 1992.
[92] Bowen, R., Hausdorff dimension of quasi-circles, Inst. Hautes Études Sci. Publ., 50, 11-26 (1979) · Zbl 0439.30032
[93] Bers, L., On Hilbert’s 22nd problem, Proc. Sympos. Pure Math., 28, 559-609 (1976) · Zbl 0348.30013
[94] Abikoff, W.; Maskit, B., Geometric decomposition of Kleinian groups, Amer. J. Math., 99, 687-697 (1977) · Zbl 0374.30018
[95] J. Polchinski, String Theory, Vols. I and II, Cambridge University Press, Cambridge, 1998.; J. Polchinski, String Theory, Vols. I and II, Cambridge University Press, Cambridge, 1998. · Zbl 1006.81522
[96] L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York, 1966.; L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York, 1966. · Zbl 0138.06002
[97] Gardiner, F.; Sullivan, D., Symmetric structures on a closed curve, Amer. J. Math., 114, 683-736 (1992) · Zbl 0778.30045
[98] Beurling, A.; Ahlfors, L., The boundary correspondence under quasiconformal mappings, Acta Math., 96, 124-142 (1956) · Zbl 0072.29602
[99] Carleson, L., On mappings conformal at the boundary, J. Analyse Math., 19, 1-13 (1967) · Zbl 0186.13701
[100] Agard, S.; Kelingos, J., On parametric representation of quasisymmetric functions, Comment. Math. Helv., 44, 446-456 (1969) · Zbl 0209.11201
[101] W. de Melo, S. van Strien, One-dimensional Dynamics, Springer, Berlin, 1993.; W. de Melo, S. van Strien, One-dimensional Dynamics, Springer, Berlin, 1993. · Zbl 0791.58003
[102] M. Hamermesh, Group Theory, Addison-Wesley, Reading, MA, 1964.; M. Hamermesh, Group Theory, Addison-Wesley, Reading, MA, 1964.
[103] J. de Azcarraga, J. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge University Press, Cambridge, 1998.; J. de Azcarraga, J. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics, Cambridge University Press, Cambridge, 1998. · Zbl 0997.22502
[104] Y. Choquet-Bruhat, C. DeWitt-Morette, Analysis, Manifolds and Physics, Part II, North-Holland, Amsterdam, 1989.; Y. Choquet-Bruhat, C. DeWitt-Morette, Analysis, Manifolds and Physics, Part II, North-Holland, Amsterdam, 1989. · Zbl 0682.58002
[105] Kuiper, N., Locally projective spaces of dimension one, Michigan Math. J., 2, 95-97 (1952) · Zbl 0058.16103
[106] Bers, L., Finite dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. (N.S.), 5, 131-172 (1981) · Zbl 0485.30002
[107] Zamolodchikov, A., Irreversibility of the flux of the renormalization group in a 2d field theory, JETP Lett., 43, 730-732 (1986)
[108] Zamolodchikov, A., Renormalization group and perturbation theory about fixed points in two-dimensional field theory, Soviet J. Nuclear Phys., 46, 1090-1096 (1987)
[109] Mostow, G., Quasiconformal mappings in \(n\)-space and the rigidity of hyperbolic space forms, Inst. Haudes Études Sci. Publ., 34, 53-104 (1968) · Zbl 0189.09402
[110] Donaldson, S.; Sullivan, D., Quasiconformal 4-manifolds, Acta Math., 163, 181-252 (1989) · Zbl 0704.57008
[111] Matsumoto, S., Foundations of flat conformal structures, Adv. Stud. Pure Math., 20, 167-261 (1992) · Zbl 0816.53020
[112] Gromov, M.; Lawson, H.; Thurston, W., Hyperbolic 4-manifolds and conformally flat 3-manifolds, Inst. Hautes Études Sci. Publ., 68, 27-45 (1988) · Zbl 0692.57012
[113] Kuiper, N., Hyperbolic 4-manifolds and tesselations, Inst. Hautes Études Sci. Publ., 80, 47-76 (1988) · Zbl 0692.57013
[114] Reimann, H., Invariant extension of quasiconformal deformations, Ann. Acad. Sci. Fenn., 10, 477-492 (1985) · Zbl 0592.30025
[115] Kapovich, M., On the dynamics of pseudo-Anosov homeomorphisms on representation varieties of surface groups, Ann. Acad. Sci. Fenn., 23, 83-100 (1998) · Zbl 0892.58058
[116] Henningson, M.; Skenderis, K., The holographic Weil anomaly, J. High Energy Phys., 9807, 23 (1998) · Zbl 0958.81083
[117] D. Johnson, J. Millson, Deformation spaces associated to compact hyperbolic manifolds, in: Discrete Groups in Geometry and Analysis, Birhäuser, Boston, MA, 1987.; D. Johnson, J. Millson, Deformation spaces associated to compact hyperbolic manifolds, in: Discrete Groups in Geometry and Analysis, Birhäuser, Boston, MA, 1987. · Zbl 0664.53023
[118] I. Kra, Automorphic Forms and Kleinian Groups, Benjamin, Reading, MA, 1972.; I. Kra, Automorphic Forms and Kleinian Groups, Benjamin, Reading, MA, 1972. · Zbl 0253.30015
[119] A. Borel, N. Wallach, Continuous Cohomology, Discrete Subgroups and Representations of Reductive Groups, Ann. of Math. Stud., Vol. 94, Princeton University Press, Princeton, NJ, 1980.; A. Borel, N. Wallach, Continuous Cohomology, Discrete Subgroups and Representations of Reductive Groups, Ann. of Math. Stud., Vol. 94, Princeton University Press, Princeton, NJ, 1980. · Zbl 0443.22010
[120] Eilenberg, S.; McLane, S., Cohomology theory in abstract groups, Ann. of Math., 48, 51-78 (1947) · Zbl 0029.34001
[121] Kourouniotis, C., Deformations of hyperbolic structures, Math. Proc. Cambridge Philos. Soc., 98, 247-261 (1985) · Zbl 0577.53041
[122] Balinskii, A.; Novikov, S., Poisson brackets of hydrodynamic type. Frobenius algebras and Lie algebras, Soviet Math. Dokl., 32, 228-231 (1985) · Zbl 0606.58018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.