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Capelli identity on multiparameter quantum linear groups

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Abstract

A quantum Capelli identity is given on the multiparameter quantum general linear group based on the (p ij , u)-condition. The multiparameter quantum Pfaffan of the (p ij , u)-quantum group is also introduced and the transformation under the congruent action is given. Generalization to the multiparameter hyper-Pfaffan and relationship with the quantum minors are also investigated.

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References

  1. Artin M, Schelter W, Tate J. Quantum deformations of GL n . Comm Pure Appl Math 1991, 44: 879–895

    Article  MathSciNet  MATH  Google Scholar 

  2. Brown K A, Goodearl K R. Lectures on Algebraic Quantum Groups. Boston: Birkhäuser, 2012

    MATH  Google Scholar 

  3. Dipper R, Donkin S. Quantum GLn. Proc Lond Math Soc (3), 1991, 63: 165–211

    Article  MATH  Google Scholar 

  4. Du J, Parshall B, Wang J P. Two-parameter quantum linear groups and the hyperbolic invariance of q-Schur algebras. J Lond Math Soc (2), 1991, 44: 420–436

    Article  MathSciNet  MATH  Google Scholar 

  5. Faddeev L D, Yu N, Reshetikhin N Yu, et al. Quantization of Lie groups and Lie algebras. In: Algebraic Analysis, vol. I. Boston: Academic Press, 1988, 129–139

    Article  MathSciNet  Google Scholar 

  6. Gelfand I, Gelfand S, Retakh V, et al. Quasideterminants. Adv Math, 2005, 193: 56–141

    Article  MathSciNet  MATH  Google Scholar 

  7. Hashimoto M, Hayashi T. Quantum multilinear algebra. Tohoku Math J (2), 1992, 44: 471–521

    Article  MathSciNet  MATH  Google Scholar 

  8. Hodges T J, Levasseur T, Toro M. Algebraic structure of multiparameter quantum groups. Adv Math, 1997, 126: 52–92

    Article  MathSciNet  MATH  Google Scholar 

  9. Howe R. Remarks on classical invariant theory. Trans Amer Math Soc, 1989, 313: 539–570. Erratum, Trans Amer Math Soc, 1990, 318:823

    Article  MathSciNet  MATH  Google Scholar 

  10. Jimbo M. A q-analogue of U(gl(N +1)), Hecke algebra, the Yang-Baxter equation. Lett Math Phys, 1996, 11: 247–252

    Article  MATH  Google Scholar 

  11. Jing N, Liu M. R-matrix realization of two-parameter quantum group Ur,s(gln). Commun Math Stat, 2014, 2: 211–230

    Article  MathSciNet  Google Scholar 

  12. Jing N, Zhang J. Quantum Pfaffans and hyper-Pfaffans. Adv Math, 2014, 265: 336–361

    Article  MathSciNet  Google Scholar 

  13. Jing N, Zhang J. Quantum permanents and Hafnians via Pfaffans. Lett Math Phys, 2016, 106: 1451–1464

    Article  MathSciNet  MATH  Google Scholar 

  14. Levasseur T, Stafford J T. The quantum coordinate ring of the special linear group. J Pure Appl Algebra, 1993, 86: 181–186

    Article  MathSciNet  MATH  Google Scholar 

  15. Manin Y I. Notes on quantum groups and quantum de Rham complexes, Teoret Mat Fiz, 1992, 92: 425–450. English translation in Theoret Math Phys, 1992, 92: 997–1023

    MathSciNet  MATH  Google Scholar 

  16. Molev A. Yangians and Classical Lie Algebras. Providence: Amer Math Soc, 2007

    Book  MATH  Google Scholar 

  17. Nazarov M L. Quantum Berezinian and the classical Capelli identity. Lett Math Phys, 1991, 21: 123–131

    Article  MathSciNet  MATH  Google Scholar 

  18. Noumi M, Umeda T, Wakayama M. A quantum analog of the Capelli identity and an elementary differential calculus on GLq(n). Duke J Math, 1994, 76: 567–594

    Article  MathSciNet  MATH  Google Scholar 

  19. Reshetikhin N. Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett Math Phys, 1990, 20: 331–335

    Article  MathSciNet  MATH  Google Scholar 

  20. Reshetikhin N Y, Takhtadzhyan L A, Faddeev L D. Quantization of Lie groups and Lie algebras. Algebra i Analiz, 1989, 1: 178–206. English translation in Leningrad Math J, 1990, 1: 193–225

    MathSciNet  MATH  Google Scholar 

  21. Schirrmacher A. The multiparametric deformation of GL(n) and the covariant differential calculus on the quantum vector space. Z Phys C, 1991, 50: 321–327

    Article  MathSciNet  Google Scholar 

  22. Sudbery A. Consistent multiparameter quantization of GL(n). J Phys A, 1990, 23: 697–704.

    Article  MathSciNet  MATH  Google Scholar 

  23. Takeuchi M. A two parameter quantization of GL (summary). Proc Japan Acad Ser A Math Sci, 1990, 66: 112–114

    Article  MathSciNet  MATH  Google Scholar 

  24. Weyl H. The Classical Groups, Their Invariants and Representations. Princeton: Princeton University Press, 1946

    MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11531004), Fapesp (Grant No. 2015/05927-0), Humboldt Foundation and Simons Foundation (Grant No. 523868). Jing acknowledges the support of Max-Planck Institute for Mathematics in the Sciences, Leipzig. The authors also thank South China University of Technology for support during the work.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Naihuan Jing

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Naihuan Jing

  3. Institute of Mathematics and Statistics, University of Sao Paulo, Sao Paulo, 05315-970, Brazil

    Jian Zhang

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  1. Naihuan Jing
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  2. Jian Zhang
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Correspondence to Naihuan Jing.

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Jing, N., Zhang, J. Capelli identity on multiparameter quantum linear groups. Sci. China Math. 61, 253–268 (2018). https://doi.org/10.1007/s11425-017-9216-x

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  • DOI: https://doi.org/10.1007/s11425-017-9216-x

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