Quantum groups. (English) Zbl 0641.16006
Translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 155, 18-49 (Russian) (1986; Zbl 0617.16004).
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 17B99 | Lie algebras and Lie superalgebras |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 17B35 | Universal enveloping (super)algebras |
| 81T99 | Quantum field theory; related classical field theories |
Citations:
Zbl 0617.16004References:
| [1] | J. W. Milnor and J. C. Moore, ?On the structure of Hopf algebras,? Ann. Math.,81, No. 2, 211?264 (1965). · Zbl 0163.28202 · doi:10.2307/1970615 |
| [2] | M. E. Sweedler, Hopf Algebras, Mathematical Lecture Notes Series, Benjamin, N.Y. (1969). |
| [3] | E. Abe, Hopf Algebras, Cambridge Tracts in Math., No. 74, Cambridge Univ. Press, Cambridge-New York (1980). |
| [4] | G. M. Bergman, ?Everybody knows what a Hopf algebra is,? Contemporary Mathematics,43, 25?48 (1985). · Zbl 0569.16005 · doi:10.1090/conm/043/810641 |
| [5] | M. E. Sweedler, ?Integrals for Hopf algebras,? Ann. Math.,89, No. 2, 323?335 (1969). · Zbl 0174.06903 · doi:10.2307/1970672 |
| [6] | R. G. Larson, ?Characters of Hopf algebras,? J. Algebra,17, No. 3, 352?368 (1971). · Zbl 0217.33801 · doi:10.1016/0021-8693(71)90018-4 |
| [7] | W. C. Waterhouse, ?Antipodes and group-likes in finite Hopf algebras,? J. Algebra,37, No. 2, 290?295 (1975). · Zbl 0315.16007 · doi:10.1016/0021-8693(75)90078-2 |
| [8] | D. E. Radford, ?The order of the antipode of a finite dimensional Hopf algebra is finite,? Am. J. Math.,98, No. 2, 333?355 (1976). · Zbl 0332.16007 · doi:10.2307/2373888 |
| [9] | H. Shigano, ?A correspondence between observable Hopf ideals and left coideal subalgebras,? Tsukuba J. Math.,1, 149?156 (1977). · Zbl 0408.16009 · doi:10.21099/tkbjm/1496158383 |
| [10] | E. J. Taft and R. L. Wilson, ?There exist finite-dimensional Hopf algebras with antipodes of arbitrary even order,? J. Algebra,162, No. 2, 283?291 (1980). · Zbl 0428.16007 · doi:10.1016/0021-8693(80)90181-7 |
| [11] | H. Shigano, ?On observable and strongly observable Hopf ideals,? Tsukuba J. Math.,6, No. 1, 127?150 (1982). · Zbl 0525.16004 · doi:10.21099/tkbjm/1496159455 |
| [12] | B. Pareigis, ?A noncommutative noncocommutative Hopf algebra in ?nature?,? J. Algebra,70, No. 2, 356?374 (1981). · Zbl 0463.18003 · doi:10.1016/0021-8693(81)90224-6 |
| [13] | G. I. Kats, ?Generalization of the group duality principle,? Dokl. Akad. Nauk SSSR,138, No. 2, 275?278 (1961). |
| [14] | G. I. Kats, ?Compact and discrete ring groups,? Ukr. Mat. Zh.,14, No. 3, 260?270 (1962). · Zbl 0287.22002 · doi:10.1007/BF02526635 |
| [15] | G. I. Kats, ?Ring groups and duality principles,? Tr. Moskov. Mat. Obshch.,12, 259?303 (1963);13, 84?113 (1965). · Zbl 0144.37902 |
| [16] | G. I. Kats and V. G. Palyutkin, ?Example of a ring group generated by Lie groups,? Ukr. Mat. Zh.,16, No. 1, 99?105 (1964). |
| [17] | G. I. Kats and V. G. Palyutkin, ?Finite ring groups,? Tr. Moskov. Mat. Obshch.,15, 224?261 (1966). |
| [18] | G. I. Kats, ?Extensions of groups which are ring groups,? Mat. Sb.,76, No. 3, 473?496 (1968). |
| [19] | G. I. Kats, ?Some arithmetic properties of ring groups,? Funkts. Anal. Prilozhen.,6, No. 2, 88?90 (1972). |
| [20] | L. I. Vainerman and G. I. Kats, ?Nonunimodular ring groups and Hopf-von Neumann algebras,? Mat. Sb.,94, No. 2, 194?225 (1974). |
| [21] | M. Enock and J.-M. Schwartz, ?Une dualite dans les algebres de von Neumann,? Bull. Soc. Mat. France, Mem. No. 44 (1975). |
| [22] | J.-M. Schwartz, ?Relations entre ?ring groups? et algebres de Kac,? Bull. Sci. Math. (2),100, No. 4, 289?300 (1976). |
| [23] | M. Enock, ?Produit croise d’une algebre de von Neumann par une algebre de Kac,? J. Funct. Analysis,26, No. 1, 16?47 (1977). · Zbl 0366.46053 · doi:10.1016/0022-1236(77)90014-3 |
| [24] | M. Enock and J.-M. Schwartz, ?Produit croise d’une algebre de von Neumann par une algebre de Kac, II,? Publ. RIMS Kyoto Univ.,16, No. 1, 198?232 (1980). · Zbl 0441.46056 · doi:10.2977/prims/1195187504 |
| [25] | J. de Canniere, M. Enock, and J.-M. Schwartz, ?Algebres de Fourier associees a une algebre de Kac,? Math. Ann.,245, No. 1, 1?22 (1979). · Zbl 0402.43001 · doi:10.1007/BF01420426 |
| [26] | J. de Canniere, M. Enock, and J.-M. Schwartz, ?Sur deux resultats d’analyse harmonique non-commutative: Une application de la theorie des algebres de Kac,? J. Operator Theory,5, No. 2, 171?194 (1981). · Zbl 0486.22003 |
| [27] | V. G. Drinfel’d, ?Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the classical Yang-Baxter equations,? Dokl. Akad. Nauk SSSR,268, No. 2, 285?287 (1982). |
| [28] | V. G. Drinfel’d, ?Hopf algebras and the quantum Yang-Baxter equation,? Dokl. Akad. Nauk SSSR,283, No. 5, 1060?1064 (1985). |
| [29] | I. M. Gel’fand and I. V. Cherednik, ?Abstract Hamiltonian formalism for classical Yang-Baxter bundles,? Usp. Mat. Nauk,38, No. 3, 3?21 (1983). · Zbl 0536.58006 |
| [30] | I. V. Cherednik, ?Bäcklund-Darboux transformations for classical Yang-Baxter bundles,? Funkts. Anal. Prilozhen.,17, No. 2, 88?89 (1983). · Zbl 0524.58017 |
| [31] | I. V. Cherednik, ?Definition of ?-functions for generalized affine Lie algebras,? Funkts. Anal. Prilozhen.,17, No. 3, 93?95 (1983). · Zbl 0528.17004 |
| [32] | I. M. Gel’fand and I. Ya. Dorfman, ?Hamiltonian operators and the classical Yang-Baxter equations,? Funkts. Anal. Prilozhen.,16, No. 4, 1?9 (1982). · Zbl 0498.32010 · doi:10.1007/BF01081801 |
| [33] | A. Weinstein, ?The local structure of Poisson manifolds,? J. Diff. Geometry,18, No. 3, 523?557 (1983). · Zbl 0524.58011 · doi:10.4310/jdg/1214437787 |
| [34] | J. A. Schouten, ?Über differentialkomitanten zweier kontravarianter Grössen,? Nederl. Akad. Wetensch. Proc. Ser. A,43, 449?452 (1940). |
| [35] | L. D. Faddeev and L. A. Takhtajan, Hamiltonian Approach to Soliton Theory, Springer-Verlag (to appear). |
| [36] | L. D. Faddeev, Integrable Models in (1+1)-Dimensional Quantum Field Theory (Lectures in Les Houches, 1982), Elsevier (1984). |
| [37] | E. K. Sklyanin, ?Quantum version of the method of the inverse scattering problem,? in: Differential Geometry, Lie Groups, and Mechanics. III, J. Sov. Math.,19, No. 5 (1982). · Zbl 0497.35072 |
| [38] | L. D. Faddeev and N. Yu. Reshetkhin, ?Hamiltonian structures for integrable models of field theory,? Teor. Mat. Fiz.,56, No. 3, 323?343 (1983). · Zbl 0541.53025 · doi:10.1007/BF01016827 |
| [39] | M. A. Semenov-Tyan-Shanskii, ?So what is the classical r-matrix,? Funkts. Anal. Prilozhen.,17, No. 4, 17?33 (1983). |
| [40] | A. A. Belavin and V. G. Drinfel’d, ?Solutions of the classical Yang-Baxter equation for simple Lie algebras,? Funkts. Anal. Prilozhen.,16, No. 3, 1?29 (1982). · Zbl 0498.32010 · doi:10.1007/BF01081801 |
| [41] | A. A. Belavin and V. G. Drinfeld, ?Triangle equations and simple Lie algebras,? Sov. Sci. Rev., Sec. C, 4, 93?165 (1984). Harwood Academic Publishers, Chur (Switzerland), New York. · Zbl 0921.58073 |
| [42] | A. A. Belavin and V. G. Drinfel’d, ?Classical Yang-Baxter equation for simple Lie algebras,? Funkts. Anal. Prilozhen.,17, No. 3, 69?70 (1983). |
| [43] | M. A. Semenov-Tian-Shansky, ?Dressing transformations and Poisson group actions,? Publ. RIMS Kyoto University,21, No. 6, 1237?1260 (1985). · doi:10.2977/prims/1195178514 |
| [44] | E. K. Sklyanin, ?On complete integrability of the Landau-Lifshitz equation,? LOMI Preprint E-3-1979, Leningrad (1979). |
| [45] | P. P. Kulish and E. K. Sklyanin, ?Solutions of the Yang-Baxter equation,? in: Differential Geometry, Lie Groups, and Mechanics. III, J. Sov. Math.,19, No. 5 (1982). · Zbl 0553.58039 |
| [46] | A. A. Belavin, ?Discrete groups and integrability of quantum systems,? Funkts. Anal. Prilozhen.,14, No. 4, 18?26 (1980). |
| [47] | G. Andrews, Theory of Partitions [Russian translation], Nauka, Moscow (1982). |
| [48] | P. P. Kulish and N. Yu. Reshetkhin, ?Quantum linear problem for the sine-Gordon equation and higher representations,? in: Questions of Quantum Field Theory and Statistical Physics. 2, J. Sov. Math.,23, No. 4 (1983). |
| [49] | E. K. Sklyanin, ?An algebra generated by quadratic relations,? Usp. Mat. Nauk,40, No. 2, 214 (1985). |
| [50] | M. Jimbo, ?Quantum R-matrix for the generalized Toda system,? Preprint RIMS-506, Kyoto University, 1985. |
| [51] | M. Jimbo, ?A q-difference analogue of Ug and the Yang-Baxter equation,? Lett. Math. Phys.,10, 63?69 (1985). · Zbl 0587.17004 · doi:10.1007/BF00704588 |
| [52] | M. Jimbo, ?A q-analogue of U(gl(N+1)), Hecke algebra and the Yang-Baxter equation,? Preprint RIMS-517, Kyoto University, 1985. |
| [53] | C. N. Yang, ?Some exact results for the many-body problem in one dimension with repulsive delta-function interaction,? Phys. Rev. Lett.,19, No. 23, 1312?1314 (1967). · Zbl 0152.46301 · doi:10.1103/PhysRevLett.19.1312 |
| [54] | V. G. Drinfel’d, ?New realization of Yangians and quantized affine algebras,? Preprint FTINT AN UkrSSR, No. 30?86, Kharkov (1986). |
| [55] | V. G. Kac, Infinite-Dimensional Lie Algebras, Birkhäuser, Boston-Basel-Stuttgart (1983). |
| [56] | P. Deligne and J. S. Milne, ?Tannaka Categories,? in: Hodge Cycles and Motives [Russian translation], Mir, Moscow (1985), pp. 94?201. |
| [57] | V. V. Lyubashenko, ?Hopf algebras and symmetries,? Kievsk. Politekh. Inst., Kiev (1985), Dep. RZhMat, 1985, No. 8A447. |
| [58] | R. J. Baxter, ?Partition function of the eight-vertex lattice model,? Ann. Phys.,70, 193?228 (1972). · Zbl 0236.60070 · doi:10.1016/0003-4916(72)90335-1 |
| [59] | V. E. Korepin, ?Analysis of bilinear relations of the six-vertex model,? Dokl. Akad. Nauk SSSR,265, No. 6, 1361?1364 (1982). |
| [60] | V. O. Tarasov, ?Structure of quantized L-operators for the L matrix of the XXZ-model,? Teor. Mat. Fiz.,61, No. 2, 163?173 (1984). |
| [61] | V. O. Tarasov, ?Irreducible monodromy matrices for the R-matrix of the XXZ-model and lattice local quantum HamiItonians,? Teor. Mat. Fiz.,63, No. 2, 175?196 (1985). · doi:10.1007/BF01017900 |
| [62] | N. Yu. Reshetikhin, ?Integrable models of quantum one-dimensional magnetics with O(n) and Sp(2k)-symmetries,? Teor. Mat. Fiz.,63, No. 3, 347?366 (1985). |
| [63] | P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin, ?Yang-Baxter equation and representation theory. I,? Lett. Math. Phys.,5, No. 5, 393?403 (1985). · Zbl 0502.35074 · doi:10.1007/BF02285311 |
| [64] | V. G. Drinfel’d, ?Degenerate affine Hecke algebras and Yangians,? Funkts. Anal. Prilozhen.,20, No. 1, 69?70 (1986). |
| [65] | L. A. Takhtadzhyan, ?Quantum method of the inverse problem and algebraized matrix Bethe ansatz,? in: Questions of Quantum Field Theory and Statistical Physics, J. Sov. Math.,23, No. 4 (1983). |
| [66] | P. P. Kulish and N. Yu. Reshetikhin, ?Generalized Heisenberg ferromagnetics and the Gross-Neveu model,? Zh. Eksp. Teor. Fiz.,80, No. 1, 214?227 (1981). |
| [67] | N. Yu. Reshetikhin, ?Method of functional equations in the theory of precisely solvable quantum systems,? Zh. Eksp. Teor. Fiz.,84, No. 3, 1190?1201 (1983). |
| [68] | N. Yu. Reshetikhin, ?Exactly solvable quantum mechanical systems on a lattice, connected with classical Lie algebras,? in: Differential Geometry, Lie Groups, and Mechanics. V. J. Sov. Math.,28, No. 4 (1985). |
| [69] | E. K. Sklyanin and L. D. Faddeev, ?Quantum-mechanical approach to completely integrable models of field theory,? Dokl. Akad. Nauk SSSR,243, No. 6, 1430?1433 (1978). |
| [70] | E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, ?Quantum method of the inverse problem. I,? Teor. Mat. Fiz.,40, No. 2, 194?220 (1979). · Zbl 1138.37331 · doi:10.1007/BF01018718 |
| [71] | L. A. Takhtadzhyan and L. D. Faddeev, ?Quantum method of the inverse problem and Heisenberg’s XX-model,? Usp. Mat. Nauk,34, No. 5, 13?63 (1979). |
| [72] | L. D. Faddeev, ?Quantum completely integrable models in field theory,? Sov. Sci. Rev., Sec. C,1, 107?155 (1980). · Zbl 0569.35064 |
| [73] | P. P. Kulish and E. K. Sklyanin, ?Quantum spectral transform method. Recent developments,? in: Integrable Quantum Field Theories (Lecture Notes in Physics, Vol. 151), Springer-Verlag, Berlin-Heidelberg-New York (1982), pp. 61?119. · Zbl 0734.35071 |
| [74] | I. V. Cherednik, ?A method of constructing factorized S-matrices in elementary functions,? Teor. Mat. Fiz.,43, No. 1, 117?119 (1980). · doi:10.1007/BF01018470 |
| [75] | A. V. Zamolodchikov and V. A. Fateev, ?Model factorized S-matrix and integrable Heisenberg chain with spine. I,? Yad. Fiz.,32, No. 2, 581?590 (1980). |
| [76] | A. G. Izergin and V. E. Korepin, ?The inverse scattering approach to the quantum Shabat-Mikhailov model,? Commun. Math. Phys.,79, 303?316 (1981). · Zbl 0441.35053 · doi:10.1007/BF01208496 |
| [77] | V. V. Bazhanov, ?Trigonometric solutions of triangle equations and classical Lie algebras,? Preprint IFVE 85-18, Serpukhov (1985). |
| [78] | D. I. Gurevich, ?Yang-Baxter equation and generalization of formal Lie theory,? Dokl. Akad. Nauk SSSR,288, No. 4, 797?801 (1986). · Zbl 0627.17006 |
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.
