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, 25:3 | Cite as

The field of quantum \(GL(N,\pmb {\mathbb {C}})\) in the C\(^*\)-algebraic setting

  • Kenny De CommerEmail author
  • Matthias Floré
Article
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Abstract

Given a unital \(*\)-algebra \(\mathscr {A}\) together with a suitable positive filtration of its set of irreducible bounded representations, one can construct a C\(^*\)-algebra \(A_0\) with a dense two-sided ideal \(A_c\) such that \(\mathscr {A}\) maps into the multiplier algebra of \(A_c\). When the filtration is induced from a central element in \(\mathscr {A}\), we say that \(\mathscr {A}\) is an s\(^*\)-algebra. We also introduce the notion of \(\mathscr {R}\)-algebra relative to a commutative s\(^*\)-algebra \(\mathscr {R}\), and of Hopf \(\mathscr {R}\)-algebra. We formulate conditions such that the completion of a Hopf \(\mathscr {R}\)-algebra gives rise to a continuous field of Hopf C\(^*\)-algebras over the spectrum of \(R_0\). We apply the general theory to the case of quantum \(GL(N,\mathbb {C})\) as constructed from the FRT-formalism.

Keywords

FRT quantum groups Quantized enveloping algebras Reflection equation algebra Locally compact quantum groups Deformation theory Continuous fields of C\(^*\)-algebras 

Mathematics Subject Classification

17B37 20G42 46L65 

Notes

References

  1. 1.
    Baaj, S., Julg, P.: Théorie bivariante de Kasparov et opérateurs non bornés dans les C\(^*\)-modules hilbertiens. C. R. Acad. Sci. Paris Sér. I Math. 296(21), 875–878 (1983)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Banica, T.: Representations of compact quantum groups and subfactors. J. Reine Angew. Math. 509, 167–198 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Blanchard, E.: Déformations de C\(^*\)-algèbres de Hopf. Bull. S.M.F. 124(1), 141–215 (1996)zbMATHGoogle Scholar
  4. 4.
    Baumann, P.: On the center of quantized enveloping algebras. J. Algebra 203(1), 244–260 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brown, K.A., Goodearl, K.R.: Lectures on Algebraic Quantum Groups Series: Advanced courses in mathematics. CRM Barcelona/Birkhäuser, Basel (2002)CrossRefGoogle Scholar
  6. 6.
    Parshall, B., Wang, J.-P.: Quantum Linear Groups, vol. 439. Memoirs of the American Mathematical Society, Providence (1991)zbMATHGoogle Scholar
  7. 7.
    De Commer, K., Floré, M.: A field of quantum upper triangular matrices. Int. Math. Res. Not. 2017(16), 5047–5077 (2017)MathSciNetGoogle Scholar
  8. 8.
    Donin, J., Mudrov, A.: Explicit equivariant quantization on coadjoint orbits of \(GL(n, C)\). Lett. Math. Phys. 62(1), 17–32 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Drabant, B., Schlieker, M., Weich, W., Zumino, B.: Complex quantum groups and their quantum universal enveloping algebras. Commun. Math. Phys. 147, 625–633 (1992)zbMATHCrossRefGoogle Scholar
  10. 10.
    Drinfel’d, V.G.: Quantum groups. In: Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 155 (1986). Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII193, 18–49, Translation in J. Soviet Math. 41(2) (1988), 898–915Google Scholar
  11. 11.
    Faddeev, L.D., Reshetikhin, N.Y., Takhtadzhyan, L.A.: Quantization of Lie groups and Lie algebras. Algebra Anal. 1(1), 178–206 (1989)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gerstenhaber, M., Schaps, M.: Hecke algebras, \(U_q({\mathfrak{sl}}(n))\) and the Donald–Flanigan conjecture for \(S_n\). Trans. AMS 349(8), 3353–3371 (1997)zbMATHGoogle Scholar
  13. 13.
    Hayashi, T.: Quantum deformations of classical groups. Publ. RIMS Kyoto Univ. 28, 57–81 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jordan, D., White, N.: The center of the reflection equation algebra via quantum minors. Preprint. arXiv:1709.09149v1
  15. 15.
    Joseph, A., Letzter, G.: Local finiteness of the adjoint action for quantized enveloping algebras. J. Algebra 153, 289–318 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Koelink, H.T.: On \(*\)-representations of the Hopf \(*\)-algebra associated with the quantum group \(U_q(N)\). Compos. Math. 77, 199–231 (1991)zbMATHGoogle Scholar
  17. 17.
    Klimyk, A., Schmudgen, K.: Quantum Groups and Their Representations, Texts and Monographs in Physics. Springer, Berlin (1997)zbMATHCrossRefGoogle Scholar
  18. 18.
    Kolb, S., Stokman, J.V.: Reflection equation algebras, coideal subalgebras, and their centres. Sel. Math. (N.S.) 15(4), 621–664 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Majid, S.: Examples of braided groups and braided matrices. J. Math. Phys. 32, 3061–3073 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Meyer, R.: Representations of \(*\)-algebras by unbounded operators: C\(^*\)-Hulls, local-global principle, and induction. Doc. Math. 22, 1375–1466 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Monk, A., Voigt, C.: Complex quantum groups and a deformation of the Baum–Connes assembly map. preprint, arXiv:1804.09384
  22. 22.
    Mudrov, A.: Quantum conjugacy classes of simple matrix groups. Commun. Math. Phys. 272, 635–660 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Mudrov, A.: On quantization of the Semenov–Tian–Shansky Poisson bracket on simple algebraic groups. St. Petersburg Math. J. 18, 797–808 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Nagy, G.: On the Haar measure of quantum \(SU(N)\) groups. Commun. Math. Phys. 153, 217–228 (1993)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Nagy, G.: A deformation quantization procedure for C\(^*\)-algebras. J. Oper. Theory 44, 369–411 (2000)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Neshveyev, S., Tuset, L.: K-homology class of the Dirac operator on a compact quantum group. Doc. Math. 16, 767–780 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Neshveyev, S., Tuset, L.: Compact Quantum Groups and Their Representation Categories, Cours Spécialisés [Specialized Courses], vol. 20. Société Mathématique de France, Paris (2013)zbMATHGoogle Scholar
  28. 28.
    Phillips, N.C.: Inverse limits of C\(^*\)-algebras. J. Oper. Theory 19(1), 159–195 (1988)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Podleś, P.: Complex quantum groups and their real representations. Publ. RIMS Kyoto Univ. 28, 709–745 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Pyatov, P., Saponov, P.: Characteristic relations for quantum matrices. J. Phys. A 28(15), 4415 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Rieffel, M.: Continuous fields of C\(^*\)-algebras coming from group cocycles and actions. Math. Ann. 283(4), 631–643 (1989)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Schmüdgen, K.: Über LMC\(^*\)-algebren. Math. Nachr. 68, 167–182 (1975)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Takesaki, M.: Theory of Operator Algebras. II. Springer, Berlin (2003)zbMATHGoogle Scholar
  34. 34.
    Vaes, S., Van Daele, A.: Hopf C\(^*\)-algebras. Proc. Lond. Math. Soc. 82(2), 337–384 (2001)zbMATHGoogle Scholar
  35. 35.
    Van Daele, A.: Multiplier Hopf algebras. Trans. Am. Math. Soc. 342(2), 917–932 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Van Daele, A.: An algebraic framework for group duality. Adv. Math. 140(2), 323–366 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Watatani, Y.: Index for C\(^*\)-subalgebras. Mem. Am. Math. Soc. 83(424), 1–117 (1990)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Woronowicz, S.L.: C\(^*\)-algebras generated by unbounded elements. Rev. Math. Phys. 7(3), 481–521 (1995)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Woronowicz, S.L.: Compact Quantum Groups, Symétries Quantiques (Les Houches, 1995), pp. 845–884. North-Holland, Amsterdam (1998)Google Scholar
  40. 40.
    Zakrzewski, S.: Realifications of complex quantum groups. In: Gielerak, R., et al. (eds.) Groups and Related Topics, pp. 83–100. Kluwer, Dordrecht (1992)zbMATHCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Vakgroep WiskundeVrije Universiteit Brussel (VUB)BrusselsBelgium

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