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

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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.

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References

  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)

    MathSciNet  MATH  Google Scholar 

  2. Banica, T.: Representations of compact quantum groups and subfactors. J. Reine Angew. Math. 509, 167–198 (1999)

    MathSciNet  MATH  Google Scholar 

  3. Blanchard, E.: Déformations de C\(^*\)-algèbres de Hopf. Bull. S.M.F. 124(1), 141–215 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Baumann, P.: On the center of quantized enveloping algebras. J. Algebra 203(1), 244–260 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Brown, K.A., Goodearl, K.R.: Lectures on Algebraic Quantum Groups Series: Advanced courses in mathematics. CRM Barcelona/Birkhäuser, Basel (2002)

    Google Scholar 

  6. Parshall, B., Wang, J.-P.: Quantum Linear Groups, vol. 439. Memoirs of the American Mathematical Society, Providence (1991)

    MATH  Google Scholar 

  7. De Commer, K., Floré, M.: A field of quantum upper triangular matrices. Int. Math. Res. Not. 2017(16), 5047–5077 (2017)

    MathSciNet  Google Scholar 

  8. Donin, J., Mudrov, A.: Explicit equivariant quantization on coadjoint orbits of \(GL(n, C)\). Lett. Math. Phys. 62(1), 17–32 (2002)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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), 898–915

  11. Faddeev, L.D., Reshetikhin, N.Y., Takhtadzhyan, L.A.: Quantization of Lie groups and Lie algebras. Algebra Anal. 1(1), 178–206 (1989)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  13. Hayashi, T.: Quantum deformations of classical groups. Publ. RIMS Kyoto Univ. 28, 57–81 (1992)

    MathSciNet  MATH  Google Scholar 

  14. Jordan, D., White, N.: The center of the reflection equation algebra via quantum minors. Preprint. arXiv:1709.09149v1

  15. Joseph, A., Letzter, G.: Local finiteness of the adjoint action for quantized enveloping algebras. J. Algebra 153, 289–318 (1992)

    MathSciNet  MATH  Google Scholar 

  16. Koelink, H.T.: On \(*\)-representations of the Hopf \(*\)-algebra associated with the quantum group \(U_q(N)\). Compos. Math. 77, 199–231 (1991)

    MATH  Google Scholar 

  17. Klimyk, A., Schmudgen, K.: Quantum Groups and Their Representations, Texts and Monographs in Physics. Springer, Berlin (1997)

    MATH  Google Scholar 

  18. Kolb, S., Stokman, J.V.: Reflection equation algebras, coideal subalgebras, and their centres. Sel. Math. (N.S.) 15(4), 621–664 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Majid, S.: Examples of braided groups and braided matrices. J. Math. Phys. 32, 3061–3073 (1991)

    MathSciNet  MATH  Google Scholar 

  20. Meyer, R.: Representations of \(*\)-algebras by unbounded operators: C\(^*\)-Hulls, local-global principle, and induction. Doc. Math. 22, 1375–1466 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Monk, A., Voigt, C.: Complex quantum groups and a deformation of the Baum–Connes assembly map. preprint, arXiv:1804.09384

  22. Mudrov, A.: Quantum conjugacy classes of simple matrix groups. Commun. Math. Phys. 272, 635–660 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Mudrov, A.: On quantization of the Semenov–Tian–Shansky Poisson bracket on simple algebraic groups. St. Petersburg Math. J. 18, 797–808 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Nagy, G.: On the Haar measure of quantum \(SU(N)\) groups. Commun. Math. Phys. 153, 217–228 (1993)

    MathSciNet  MATH  Google Scholar 

  25. Nagy, G.: A deformation quantization procedure for C\(^*\)-algebras. J. Oper. Theory 44, 369–411 (2000)

    MathSciNet  MATH  Google Scholar 

  26. Neshveyev, S., Tuset, L.: K-homology class of the Dirac operator on a compact quantum group. Doc. Math. 16, 767–780 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Neshveyev, S., Tuset, L.: Compact Quantum Groups and Their Representation Categories, Cours Spécialisés [Specialized Courses], vol. 20. Société Mathématique de France, Paris (2013)

    MATH  Google Scholar 

  28. Phillips, N.C.: Inverse limits of C\(^*\)-algebras. J. Oper. Theory 19(1), 159–195 (1988)

    MathSciNet  MATH  Google Scholar 

  29. Podleś, P.: Complex quantum groups and their real representations. Publ. RIMS Kyoto Univ. 28, 709–745 (1992)

    MathSciNet  MATH  Google Scholar 

  30. Pyatov, P., Saponov, P.: Characteristic relations for quantum matrices. J. Phys. A 28(15), 4415 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Rieffel, M.: Continuous fields of C\(^*\)-algebras coming from group cocycles and actions. Math. Ann. 283(4), 631–643 (1989)

    MathSciNet  MATH  Google Scholar 

  32. Schmüdgen, K.: Über LMC\(^*\)-algebren. Math. Nachr. 68, 167–182 (1975)

    MathSciNet  MATH  Google Scholar 

  33. Takesaki, M.: Theory of Operator Algebras. II. Springer, Berlin (2003)

    MATH  Google Scholar 

  34. Vaes, S., Van Daele, A.: Hopf C\(^*\)-algebras. Proc. Lond. Math. Soc. 82(2), 337–384 (2001)

    MATH  Google Scholar 

  35. Van Daele, A.: Multiplier Hopf algebras. Trans. Am. Math. Soc. 342(2), 917–932 (1994)

    MathSciNet  MATH  Google Scholar 

  36. Van Daele, A.: An algebraic framework for group duality. Adv. Math. 140(2), 323–366 (1998)

    MathSciNet  MATH  Google Scholar 

  37. Watatani, Y.: Index for C\(^*\)-subalgebras. Mem. Am. Math. Soc. 83(424), 1–117 (1990)

    MathSciNet  MATH  Google Scholar 

  38. Woronowicz, S.L.: C\(^*\)-algebras generated by unbounded elements. Rev. Math. Phys. 7(3), 481–521 (1995)

    MathSciNet  MATH  Google Scholar 

  39. Woronowicz, S.L.: Compact Quantum Groups, Symétries Quantiques (Les Houches, 1995), pp. 845–884. North-Holland, Amsterdam (1998)

    Google Scholar 

  40. Zakrzewski, S.: Realifications of complex quantum groups. In: Gielerak, R., et al. (eds.) Groups and Related Topics, pp. 83–100. Kluwer, Dordrecht (1992)

    MATH  Google Scholar 

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  1. Vakgroep Wiskunde, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050, Brussels, Belgium

    Kenny De Commer & Matthias Floré

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  1. Kenny De Commer
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  2. Matthias Floré
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Correspondence to Kenny De Commer.

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Dedicated to the memory of Étienne Blanchard.

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The work of K. De Commer was supported by the FWO Grant G.0251.15N, and this work is part of the project supported by the Grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS.

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De Commer, K., Floré, M. The field of quantum \(GL(N,\pmb {\mathbb {C}})\) in the C\(^*\)-algebraic setting. Sel. Math. New Ser. 25, 3 (2019). https://doi.org/10.1007/s00029-019-0456-0

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