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Twisted Quantum Toroidal Algebras \({T_q^-(\mathfrak g)}\)

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Abstract

We construct a principally graded quantum loop algebra for the Kac–Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.

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References

  1. Feigin B., Jimbo M., Miwa T., Mukhin E.: Quantum continuous \({\mathfrak{gl}_\infty}\): Semi-infinite construction of representations. Kyoto J. Math. 51, 337–364 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Feigin B., Jimbo M., Miwa T., Mukhin E.: Quantum toroidal \({\mathfrak{gl}_1}\)-algebra: plane partitions. Kyoto J. Math. 52(3), 621–659 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Feigin B., Jimbo M., Miwa T., Mukhin E.: Representations of quantum toroidal \({\mathfrak{gl}_n}\). J. Algebra 380, 78–108 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Frenkel, I.B.: Representation of Kac–Moody algebras and dual resonance models. In: Applications of group theory in physics and mathematical physics, Chicago, 1982. Lecture Notes in Applied Mathematics, pp. 325–353. American Mathematical Society, Providence (1985)

  5. Frenkel I.B., Jing N.: Vertex representations of quantum affine algebras. Proc. Natl. Acad. Sci. USA 85, 9373–9377 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Frenkel I.B., Jing N., Wang W.: Quantum vertex representations via finite groups and the Mckay correspondence. Commun. Math. Phys. 211, 365–393 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Frenkel I.B., Lepowsky J., Meurman A.: Vertex operator algebras and the Monster. Academic Press, San Diego (1988)

    MATH  Google Scholar 

  8. Gao Y., Jing N.: \({U_{q}(\widehat{gl}_{N})}\) action on \({\widehat{gl}_{N}}\)-modules and quantum toroidal algebras. J. Algebra 273, 320–343 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gao Y., Jing N.: A quantized Tits–Kantor–Koecher algebra. Algebra Represent. Theory 13, 207–217 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ginzburg V., Kapranov M., Vasserot E.: Langlands reciprocity for algebraic surfaces. Math. Res. Lett. 2, 147–160 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hernandez D.: Representations of quantum affinizations and fusion product. Transform. Groups 10, 163–200 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hernandez D.: Quantum toroidal algebras and their representations. Selecta Math. (N.S.) 14, 701–725 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jing N.: Twisted vertex representations of quantum affine algebras. Invent. Math. 102, 663–690 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Jing N.: Quantum Kac–Moody algebras and vertex representations. Lett. Math. Phys. 44, 261–271 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jing, N.: New twisted quantum current algebras. In: Wang, J., Lin, Z. (eds.) Representations and quantizations, pp. 263–274. Higher Ed. Press, Beijing (2000)

  16. Jing N., Liu R.: A twisted quantum toroidal algebra. Front. Math. China 5, 1117–1128 (2013)

    Article  MathSciNet  Google Scholar 

  17. Lepowsky J., Wilson R.L.: Construction of the affine Lie algebra \({A^{(1)}_1}\). Commun. Math. Phys. 62, 43–53 (1978)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Miki K.: Representations of quantum toroidal algebra \({U_q(sl_{n+1,tor}) (n\geq 2)}\). J. Math. Phys. 41, 7079–7098 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Moody R.V., Rao S.E., Yokonuma T.: Toroidal Lie algebras and vertex representations. Geom. Dedicata 35, 283–307 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Nakajima, H.: Geometric construction of representations of affine algebras. In: Proceedings of the ICM, vol. I, pp. 423–438. Higher Ed. Press, Beijing (2002)

  21. Saito Y.: Quantum toroidal algebras and their vertex representations. Publ. RIMS Kyoto Univ. 34, 155–177 (1998)

    Article  MATH  Google Scholar 

  22. Saito Y., Takemura K., Uglov D.: Toroidal actions on level-1 modules of \({U_{q}(\widehat{sl_{n}})}\). Transform. Groups 3, 75–102 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Takemura K., Uglov D.: Representations of the quantum toroidal algebra on highest weight modules of the quantum affine algebra of type \({gl_{N}}\). Publ. RIMS Kyoto Univ. 35, 407–450 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Varagnolo M., Vasserot E.: Schur duality in the toroidal setting. Commun. Math. Phys. 182, 469–484 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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Correspondence to Rongjia Liu.

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Jing, N., Liu, R. Twisted Quantum Toroidal Algebras \({T_q^-(\mathfrak g)}\) . Lett Math Phys 104, 1137–1145 (2014). https://doi.org/10.1007/s11005-014-0711-4

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  • DOI: https://doi.org/10.1007/s11005-014-0711-4

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