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Quantum Groups: An Introduction and Survey for Ring Theorists

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Noncommutative Rings

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 24))

Abstract

This article describes recent work on new classes of non-commutative algebras which have been dubbed quantum algebras, quantum groups, and quantized enveloping algebras. The basic ring theoretic properties are described, and a number of questions and problems are raised.

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References

  1. H.H. Andersen, The Linkage Principle and the sum formula for quantum groups. Preprint, 1989.

    Google Scholar 

  2. H. H. Anderson and P. Polo Representations of quantum algebras. Preprint, 1990.

    Google Scholar 

  3. M. Artin and W. Schelter Graded algebras of global dimension 3n Adv. Math. 66 (1987), 171–216.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Artin, J. Tate and M. van den Bergh Some algebras associated to automorphisms of curves. Preprint, 1989.

    Google Scholar 

  5. M. Artin, J. Tate and M. van den Bergh Modules over regular algebras of dimension 3. Preprint, 1989.

    Google Scholar 

  6. M. Artin and M. van den Bergh Twisted Homogeneous Coordinate Rings. Preprint, 1990.

    Google Scholar 

  7. J. Backelin and R. Froberg Koszul algebras, Veronese subrings and rings with linear resolution Revue Roumaine de Math. Pures et Appl. 30 (1985), 85–97.

    MATH  MathSciNet  Google Scholar 

  8. R. J. Baxter, “Exactly solved models in statistical mechanics,” Academic Press, New York, 1982.

    MATH  Google Scholar 

  9. A. Beilinson and V. Ginsburg, Mixed categories, Ext-duality and Representations (results and conjectures). Preprint.

    Google Scholar 

  10. A. A. Belavin Discrete Groups and the Integrability of Quantum Systems Func. Anal, and its Appl. 14 (1980), 260–267.

    Article  MathSciNet  Google Scholar 

  11. A. A. Belavin and V. G. Drinfeld On the solutions of the classical Yang- Baxter equations, Func. Anal, and Appl. 16 (1982), 159–180.

    Article  MathSciNet  Google Scholar 

  12. A. A. Belavin and V. G. Drinfeld Triangle Equations and simple Lie algebras, Sov. Sci. Rev. C4 (1984).

    Google Scholar 

  13. G. Bergman Diamond Lemma for Ring Theory, Adv. Math. 29 (1978), 178–218.

    Article  MathSciNet  Google Scholar 

  14. K. Bragiel Twisted SU(3) group to appear.

    Google Scholar 

  15. I. V. Cherednik Some finite dimensional representations of generalized Sklyanin algebras, Func. Anal, and Appl 19 (1985), 77–79.

    Google Scholar 

  16. R. Dipper and S. Donkin Quantum GL. Preprint.

    Google Scholar 

  17. P. Doubilet and G. C. Rota Skew-symmetric Invariant Theory Adv. Math. 21 (1976), 196–203.

    Article  MATH  MathSciNet  Google Scholar 

  18. V. G. Drinfeld Hamiltonian structures on Lie groups, Lie bialgebras, and the geometric meaning of the Yang-Baxter equations, Sov. Math. Dokl. 32 (1985), 254–258.

    Google Scholar 

  19. V. G. Drinfeld Hopf algebras and the quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985), 254–258.

    Google Scholar 

  20. V. G. Drinfeld Degenerate affine Hecke algebras and Yangians, Func. Anal, and Appl. 20 (1986), 58–60.

    Article  MathSciNet  Google Scholar 

  21. V. G. Drinfeld Quantum Groups Proc. Int. Cong. Math.; Berkeley, 1 (1986), 798–820.

    Google Scholar 

  22. V. G. Drinfeld A new realisation of Yangians and quantised affine algebras, Sov. Math. Dokl. 36/296 (1988), 212–216.

    MathSciNet  Google Scholar 

  23. V. G. Drinfeld On quadratic commutation relations in the quasi-classic limit, Mat. Fizika Funkc. Analiz. Kiev, 25–33. (in Russian).

    Google Scholar 

  24. J. Du, B. Parshall and J-P. Wang, Two parameter quantum linear groups and the Hyperbolic invariance of q-Schur algebra. Preprint, 1990.

    Google Scholar 

  25. L. D. Faddeev, N. Y. Reshetikhin and L. A. Takhtajan Quantization of Lie groups and Lie algebras. Preprint LOMI

    Google Scholar 

  26. L. D. Faddeev, and L. A. Takhtajan Liouville model on the lattice, in “Lecture Notes in Physics, No. 246,” Springer-Verlag, 1986, pp. 166–178.

    Google Scholar 

  27. M. Gerstenhaber On the deformation of rings and algebras Ann. Math. 79 (1964), 59–103.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. Gerstenhaber and S. D. Schack Quantum groups as deformations of Hopf algebras Proc. Nat. Acad. Sci. 87 (1990), 478–481.

    Article  MATH  MathSciNet  Google Scholar 

  29. T. Hayashi Q-analogues of Clifford and Weyl algebras. Spinor and oscillator representations of quantum enveloping algebras. Preprint, Nagoya, 1990.

    Google Scholar 

  30. M. Jimbo A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Mat. Phys. 10 (1985), 63–69.

    Article  MATH  MathSciNet  Google Scholar 

  31. M. Jimbo, Quantum R matrix for the generalised Toda system, Commun. Math. Phys. 102 (1986), 537–547.

    Article  MATH  Google Scholar 

  32. M. Jimbo A q-analogue of U(&l(n + 1)), Hecke algebra and the Yang-Baxter equation Lett. Math. Phys. 11 (1986), 247–252.

    Article  MATH  MathSciNet  Google Scholar 

  33. Naihuan Jing, Mo-Lin Ge, Yong-Shi Wu New quantum group associated with a “non-standard” Braid group representation. Preprint, 1990.

    Google Scholar 

  34. V. F. R. Jones Polynomial invariants of knots via von Neumann algebras Bull. Amer. Math. Soc. 12 (1985), 103–111.

    Article  MATH  MathSciNet  Google Scholar 

  35. V. F. R. Jones Hecke algebra representations of braid groups, and link polynomials, Ann. Math. 126 (1987), 335–388.

    Article  MATH  Google Scholar 

  36. V. F. R. JonesOn knot invariants related to some statistical mechanical models.

    Google Scholar 

  37. A. A. Kirillov and N. Yu. ReshetikhinThe Yangians, Bethe Ansatz and combinatorics Lett. Math. Phys. 12 (1986), 199–208.

    MATH  Google Scholar 

  38. A. A. Kirillov and N. Yu. ReshetikhinRepresentations of the algebra U q (sl(2)), q-orthogonal polynomials and invariants of links. LOMI preprint (1988).

    Google Scholar 

  39. T. KohnoMonodromy representations of Braid groups and Yang-Baxter equations, Ann. Inst. Fourier 37 (1987), 139–160.

    Article  MATH  MathSciNet  Google Scholar 

  40. Y. Kosmann-SchwarzbachPoisson-Drinfeld groups; Proc. Oberwolfach Conf. on non-linear evolution equations; M. Albowitz, B. Fuchsteiner, M. Kruskal ed. World Scientific Publ. (1986).

    Google Scholar 

  41. P. P. Kulish and N. ReshetikhinQuantum linear problem for the Sine-Gordon equation and higher representations, J. Sov. Math. 23 (1983), 2435–2441.

    Article  Google Scholar 

  42. P. P. Kulish, N. Reshetikhin, and E. K. SklyaninYang-Baxter equation and representation theory Lett. Math. Phys. 5 (1981), 393–403.

    Article  MATH  MathSciNet  Google Scholar 

  43. P. P. Kulish and E. K. SklyaninSolutions of the Yang-Baxter equation J. Sov. Math. 19 (1982), 1596–1620.

    Article  MATH  Google Scholar 

  44. C. LofwallOn the subalgebra generated by the 1-dimensional elements in the Yoneda Ext-algebra, Springer LNM 1183 (1988), 291–338.

    MathSciNet  Google Scholar 

  45. G. LusztigQuantum deformations of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), 237–249.

    Article  MATH  MathSciNet  Google Scholar 

  46. G. LusztigModular representations and quantum groups, Contemp. Math. 82 (1989), 59–78.

    MathSciNet  Google Scholar 

  47. G. Lusztig, Finite dimensional Hopf algebras arising from quantum groups. Preprint.

    Google Scholar 

  48. G. Lusztig, Quantum groups at roots of 1. Preprint.

    Google Scholar 

  49. G. LusztigCanonical bases arising from quantized enveloping algebras. Preprint, 1990.

    Google Scholar 

  50. S. MajidQuasitriangular Hopf Algebras and Yang-Baxter equations, Inter. J. Mod. Phys. 5 (1990), 1–91.

    Article  MATH  MathSciNet  Google Scholar 

  51. S. Majid and Ya. S. SoibelmanRank of quantized enveloping algebras and modular functions. Preprint, 1990

    Google Scholar 

  52. Yu. I. ManinSome remarks on Koszul algebras and Quantum groups Ann. Inst. Fourier 37 (1987), 191–205.

    Article  MATH  MathSciNet  Google Scholar 

  53. Yu. I. Manin, “Quantum groups and Non-commutative geometry,” Les Publ, du Centre de Récherches Math., Université de Montreal, 1988.

    Google Scholar 

  54. T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, M. UenoRepresentations of Quantum groups and a q-analogue of orthogonal polynomials, C. R. Acad. Sci. Paris 307 (1988), 559–564.

    MATH  MathSciNet  Google Scholar 

  55. J. C. Monnell and J. J. PettitCrossed products and multiplicative analogues of Weyl algebras J. Lond. Math. Soc. 38 (1988), 47–55.

    Article  Google Scholar 

  56. S. Montgomery and S. P. SmithSkew derivations and U q (sl(2)) Israel J. Math., to appear.

    Google Scholar 

  57. M. Noumi, H. Yamada and K. MimachiFinite dimensional representations of the quantum group GL q (n + 1,C) and the zonal spherical functions on U q (n)\U q (n + 1). Preprint, 1989.

    Google Scholar 

  58. A. V. Odesskii and B. L. FeiginSklyanin algebras associated with an elliptic curve. Preprint (1988).

    Google Scholar 

  59. A. V. Odesskii and. B. L. FeiginElliptic Sklyanin Algebras Fune. Anal. Appl. 23 (1989), 45–54.

    MATH  MathSciNet  Google Scholar 

  60. G. I. OlshanskiiYangians and universal enveloping algebras LOMI 164 (1987), 142–150.

    Google Scholar 

  61. B. Parshall and J-P. WangQuantum Linear Groups I. Preprint. University of Virginia (1989).

    Google Scholar 

  62. B. Parshall and J-P. WangQuantum Linear Groups II. Preprint.University of Virginia (1989).

    Google Scholar 

  63. P. PodlesQuantum Spheres, Lett. Math. Phys. 14 (1987), 193–202.

    Article  MATH  MathSciNet  Google Scholar 

  64. S. B. PriddyKoszul resolutions Trans. Amer. Math. Soc. 152 (1970), 39–60.

    Article  MATH  MathSciNet  Google Scholar 

  65. N. Reshetikhin, Theoret. Math. Phys. 63 (1985).

    Google Scholar 

  66. N. ReshetikhinQuantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I., LOMI (1988,), E-4–87. Preprint. II., LOMI (1988), E-17–87. Preprint.

    Google Scholar 

  67. N. Reshetikhin and V. G. TuraevInvariants of 3-manifolds via link polynomials and quantum groups, MSRI (1989). Preprint.

    Google Scholar 

  68. C. M. RingelHall Algebras and Quantum Groups. Preprint. Bielefeld (1989).

    Google Scholar 

  69. M. RossoComparaison des groupes SU(2) quantiques de Drinfeld et de Woronowiez, C. R. Acad. Sci. Paris 304 (1987), 323–326.

    MATH  MathSciNet  Google Scholar 

  70. M. RossoRepresentations irreducibles de dimension finie du q-analogue de V algebre enveloppante d’une algebre de Lie semisimple, C. R. Acad. Sci. Paris 305, 587–590.

    Google Scholar 

  71. M. RossoFinite dimensional representations of the quantum analog of the enveloping algebra of a complex semisimple Lie algebra, Comm. Math. Phys. 117, 581–593.

    Google Scholar 

  72. M. RossoGroupes quantiques et modeles à vertex de V. Jones en theorie des noeuds C. R. Acad.Sci. Paris 307 (1988), 207–210.

    MATH  MathSciNet  Google Scholar 

  73. M. RossoAn analogue of the PBW theorem and the universal R-matrix for U h (sl(n + 1)). Preprint. Palaiseau (1989).

    Google Scholar 

  74. M. RossoAnalogues de la forme de Killing et du théorème d’Harish-Chandra pour les groupes quantiques. Preprint. Palaiseau (1989).

    Google Scholar 

  75. A. N. Rudakov and I. R. ShafarevichIrreducible representations of a simple three dimensional Lie algebra over a field of finite characteristic, Math. Notes 2 (1968), 760–767.

    Google Scholar 

  76. E. K. SklyaninSome algebraic structures connected with the Yang-Bctxter equation, Fune. Anal, and Appl. 16 (1982), 263–270.

    Article  MathSciNet  Google Scholar 

  77. E. K. SklyaninSome algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebras, Func. Anal, and Appl. 17 (1983), 273–284.

    Article  MATH  MathSciNet  Google Scholar 

  78. E. K. SklyaninAn algebra generated by quadratic relations, Usp. Mat. Nauk. 40 (1985), 214.

    MathSciNet  Google Scholar 

  79. S. P. Smith and J. T. StaffordRegularity of the 4-dimensional Sklyanin algebra. in preparation.

    Google Scholar 

  80. Ya. S. Soibelman, Irreducible representations of the functional algebra of quantised SU(n) and the Schubert cells, Func. Anal, and Appl. (to appear).

    Google Scholar 

  81. E. Taft and J. TowberQuantum deformation of flag schemes and Grassman schemes I, a q-deformation of the shape algebra for GL(n). Preprint (1989).

    Google Scholar 

  82. M. TakeuchiQuantum Orthogonal and Symplectic Groups and their embedding into Quantum GL(n), Proc. Japan Acad. 65 (1989), 55–58.

    Article  MATH  Google Scholar 

  83. M. TakeuchiMatrix Bialgebras and Quantum Groups. Preprint. Tsukuba (1989).

    Google Scholar 

  84. M. Takeuchi, A two parameter quanitization of GL(n); Summary.

    Google Scholar 

  85. L. A. TakhtajanQuantum Groups and Integrable Models. Preprint. Steklov Inst. (1988).

    Google Scholar 

  86. L. A. TakhtajanNoncommutative homology of quantum tori, Func. Anal. Appl. 23 (1989), 147–149.

    Article  MathSciNet  Google Scholar 

  87. T. TanisakiHarish-Chandra isomorphisms for Quantum algebras. Preprint, Osaka Univ. (1989).

    Google Scholar 

  88. T. TanisakiFinite dimensional representations of quantum groups. Preprint, Osaka Univ. (1989).

    Google Scholar 

  89. V. G. TuraevThe Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988), 527–553.

    Article  MATH  MathSciNet  Google Scholar 

  90. L. L. Vaksman and S. Ya SoibelmanThe algebra of functions on quantised SU(2), Func. Anal, and Appl. 22 (1988), 170–181.

    Article  MATH  Google Scholar 

  91. J-L. VerdierGroupes Quantiques Seminaire Bourbaki (1986–87,). No. 685.

    Google Scholar 

  92. H. WenzlBraid group representations and the quantum Yang-Baxter equation. Preprint.

    Google Scholar 

  93. S. L. WoronowiczTwisted SU(2)-group. An example of a non-commutative differential calculus, Publ. R.I.M.S., Kyoto Univ. 23 (1987), 117–181.

    Google Scholar 

  94. S. L. WoronowiczCompact Matrix pseudogroups, Comm. Math. Phys. III (1987), 613–615.

    Article  MathSciNet  Google Scholar 

  95. S. L. WoronowiczTannaka-Krein duality for compact Matrix pseudogroups, Invent. Math. 93 (1988), 35–76.

    Article  MATH  MathSciNet  Google Scholar 

  96. Xi, NanhuaRepresentations of Finite Dimensional Hopf algebras arising from Quantum Groups. Preprint (1989).

    Google Scholar 

  97. H. YamaneA Poincare-Birkhoff-Witt Theorem for the quantum group of type A, Proc. Japan Acad. 64 (1988), 385–386.

    Article  MATH  MathSciNet  Google Scholar 

  98. H. YamaneA PBW Theorem for quantized universal enveloping algebras of type A N Publ. RIMS Kyoto 25 (1989), 503–520.

    Article  MATH  MathSciNet  Google Scholar 

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Smith, S.P. (1992). Quantum Groups: An Introduction and Survey for Ring Theorists. In: Montgomery, S., Small, L. (eds) Noncommutative Rings. Mathematical Sciences Research Institute Publications, vol 24. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9736-6_6

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