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Generalized Verma Modules over \(U_{q}(\mathfrak {sl}_{n}(\mathbb {C}))\)

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Abstract

We construct realizations of quantum generalized Verma modules for \(U_{q}(\mathfrak {sl}_{n}(\mathbb {C}))\) by quantum differential operators. Taking the classical limit \(q \rightarrow 1\) provides a realization of classical generalized Verma modules for \(\mathfrak {sl}_{n}(\mathbb {C})\) by differential operators.

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References

  1. Boe, B., Collingwood, D.: Multiplicity free categories of highest weight representations, I. Comm. Algebra 18(4), 947–1032 (1990)

    Article  MathSciNet  Google Scholar 

  2. Coleman, J.A., Futorny, V.: Stratified L-modules. J. Algebra 163(1), 219–234 (1994)

    Article  MathSciNet  Google Scholar 

  3. Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  4. Dimitrov, I., Mathieu, O., Penkov, I.: On the structure of weight modules. Trans. Amer. Math. Soc. 352(6), 2857–2869 (2000)

    Article  MathSciNet  Google Scholar 

  5. Yuri, A., Drozd, S.O., Futorny, V.: The Harish-Chandra S-homomorphism and \(\mathfrak {G}\)-modules generated by primitive elements. Ukrainian Math. J. 42(8), 919–924 (1990)

    Article  MathSciNet  Google Scholar 

  6. Fernando, S.L.: Lie algebra modules with finite-dimensional weight spaces, I. Trans. Amer. Math. Soc. 322(2), 757–781 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Futorny, V.: A generalization of Verma modules and irreducible representations of the lie algebra \(\mathfrak {sl}(3)\). Ukrain. Mat. Zh. 38(4), 492–497 (1986)

    MathSciNet  Google Scholar 

  8. Futorny, V.: The weight representations of semisimple finite dimensional lie algebras. Ph.D. thesis Kiev University (1987)

  9. Garland, H., Lepowsky, J.: Lie algebra homology and the Macdonald-Kac formulas. Invent. Math. 34(1), 37–76 (1976)

    Article  MathSciNet  Google Scholar 

  10. Humphreys, J.E.: Representations of Semisimple Lie Algebras in the BGG Category \(\mathcal {O}\) Graduate Studies in Mathematics, vol. 94. American Mathematical Society, Providence (2008)

    Book  Google Scholar 

  11. Irving, R.S., Shelton, B.: Loewy series and simple projective modules in the category \(\mathcal {O}_{S}\). Pacific J. Math. 2, 319–342 (1988)

    Article  Google Scholar 

  12. Jimbo, M.: A q-analogue of \(u(\mathfrak {gl}(n + 1))\), Hecke algebra, and the Yang–Baxter equation. Lett. Math. Phys. 11(3), 247–252 (1986)

    Article  MathSciNet  Google Scholar 

  13. Kassel, C.: Qantum groups, Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)

    Google Scholar 

  14. Klimyk, A., Schmüdgen, K.: Qantum Groups and their Representations. Springer, Berlin (1997)

    Book  Google Scholar 

  15. Lepowsky, J.: A generalization of the Bernstein-Gelfand-Gelfand resolution. J. Algebra 2, 496–511 (1977)

    Article  MathSciNet  Google Scholar 

  16. Lepowsky, J.: Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism. J. Algebra. 2, 470–495 (1977)

    Article  MathSciNet  Google Scholar 

  17. Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. A. Math. 2, 237–249 (1988)

    Article  MathSciNet  Google Scholar 

  18. Mazorchuk, V.: Generalized Verma Modules, Mathematical Studies Monograph Series, vol. 8. Lviv, VNTL Publishers (2000)

    Google Scholar 

  19. Melville, D.J.: An \(\mathbb {A}\)-form technique of quantum deformations, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998). Contemporary Mathematics, vol. 248, Amer. Math. Soc., Providence, RI, pp. 359–375 (1999)

  20. Mazorchuk, V., Stroppel, C.: Categorification of (induced) cell modules and the rough structure of generalised Verma modules. A. Math. 219(4), 1363–1426 (2008)

    Article  MathSciNet  Google Scholar 

  21. Rocha-Caridi, A.: Splitting criteria for \(\mathfrak {g}\)-modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite-dimensional, irreducible \(\mathfrak {g}\)-modules. Trans. Amer. Math. Soc. 262(2), 335–366 (1980)

    MathSciNet  MATH  Google Scholar 

  22. Reshetikhin, N.Y., Takhtajan, L.A., Faddeev, L.D.: Quantization of lie groups and lie algebras. Leningrad Math. J. 1, 193–225 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Shen, G.: Graded modules of graded lie algebras of Cartan type. I. Mixed products of modules. Sci. Sinica Ser. 6, 570–581 (1986)

    MathSciNet  MATH  Google Scholar 

  24. Verma, D.-N.: Structure of certain induced representations of complex semisimple lie algebras, Ph.D. thesis, Yale University (1966)

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Acknowledgments

V. F. is supported in part by CNPq (304467/2017-0) and by Fapesp (2014/09310-5); L. K. is supported by Capes (88887.137839/2017-00) and J. Z. is supported by Fapesp (2015/05927-0).

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Correspondence to Vyacheslav Futorny.

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Presented by: Vyjayanthi Chari

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Futorny, V., Křižka, L. & Zhang, J. Generalized Verma Modules over \(U_{q}(\mathfrak {sl}_{n}(\mathbb {C}))\). Algebr Represent Theor 23, 811–832 (2020). https://doi.org/10.1007/s10468-019-09878-4

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  • DOI: https://doi.org/10.1007/s10468-019-09878-4

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