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Universal R-Matrix of Quantum Affine \({\mathfrak{gl}(1,1)}\)

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Abstract

The universal R-matrix of the quantum affine superalgebra associated to the Lie superalgebra \({\mathfrak{gl}(1,1)}\) is realized as the Casimir element of certain Hopf pairing, based on the explicit coproduct formula of all the Drinfeld loop generators.

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References

  1. Arnaudon D., Crampé N., Frappat L., Ragoucy E.: Super Yangian \({Y(\mathfrak{osp}(1|2))}\) and the universal R-matrix of its quantum double. Commun. Math. Phys. 240, 31–51 (2003)

    Article  MATH  ADS  Google Scholar 

  2. Cai J., Wang S., Wu K., Xiong C.: Universal \({\mathcal{R}}\) -matrix of the super Yangian double \({DY(\mathfrak{gl}(1|1))}\). Commun. Theor. Phys. 29, 173–176 (1998)

    Article  MathSciNet  Google Scholar 

  3. Cai, J., Wang, S., Wu, K., Zhao, W.: Drinfel’d realization of quantum affine superalgebra \({U_q(\widehat{\mathfrak{gl}(1|1)})}\). J. Phys. A: Math. Gen. 31, 1989–1994 (1998)

  4. Damiani I.: La \({\mathcal{R}}\) -matrice pour les algèbres quantiques de type affine non tordu. Ann. Sci. Ecole Norm. Sup. 31, 493–523 (1998)

    MATH  MathSciNet  Google Scholar 

  5. Damiani, I.: The R-matrix for the (twisted) quantum affine algebras. In: Representations and quantizations (Shanghai 1998), pp. 89–144. China High. Educ. Press, Beijing (2000) arXiv:1111.4085

  6. Frenkel, E., Hernandez, D.: Baxter’s relations and spectra of quantum integrable models. Duke Math. J. (to appear) arXiv:1308.3444

  7. Frenkel E., Reshetikhin N.: The q-character of representations of quantum affine algebras and deformations of \({\mathcal{W}}\) -algebras. Recent developments in quantum affine algebras and related topics. Contemp. Math. 248, 163–205 (1999)

    Article  MathSciNet  Google Scholar 

  8. Gade R.: Universal R-matrix and graded Hopf algebra structure of \({U_q(\widehat{gl}(2|2))}\). J. Phys. A: Math. Gen. 31, 4909–4925 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Ip I., Zeitlin A.: Q-operator and fusion relations for \({C_q^{(2)}(2)}\). Lett. Math. Phys. 104, 1019–1043 (2014)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Kashiwara M.: On level-zero representations of quantized affine algebras. Duke Math. J. 112, 117–175 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kang, S., Kashiwara, M., Kim, M.: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras. Preprint arXiv:1304.0323

  12. Khoroshkin S., Tolstoy V.: Universal R-matrix for quantum (super)algebras. Commun. Math. Phys. 141, 599–617 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Khoroshkin S., Tolstoy V.: The universal R-matrix for quantum untwisted affine Lie algebras. Funct. Anal. Appl. 26, 69–71 (1992)

    Article  MATH  Google Scholar 

  14. Khoroshkin S., Tolstoy V.: Yangian double. Lett. Math. Phys. 36, 373–402 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Rej A., Spill F.: The Yangian of \({\mathfrak{sl}(m|n)}\) and its quantum R-matrices. JHEP 05, 012 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  16. Zhang H.: Representations of quantum affine superalgebras. Math. Z. 278, 663–703 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhang, H.: RTT realization of quantum affine superalgebras and tensor products. Intern. Math. Res. Notes. doi:10.1093/imrn/rnv167. arXiv:1407.7001

  18. Zhang, H.: Asymptotic representations of quantum affine superalgebras. Preprint arXiv:1410.0837

  19. Zhang Y.: Comments on the Drinfeld realization of quantum affine superalgebra \({U_q(\mathfrak{gl}(m|n)^{(1)})}\) and its Hopf algebra structure. J. Phys. A: Math. Gen. 30, 8325–8335 (1997)

    Article  MATH  ADS  Google Scholar 

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  1. Université Paris Diderot-Paris 7, Institut de Mathématiques de Jussieu-Paris Rive Gauche CNRS UMR 7586, Bâtiment Sophie Germain, Case 7012, 75025, Paris Cedex 13, France

    Huafeng Zhang

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  1. Huafeng Zhang
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Zhang, H. Universal R-Matrix of Quantum Affine \({\mathfrak{gl}(1,1)}\) . Lett Math Phys 105, 1587–1603 (2015). https://doi.org/10.1007/s11005-015-0797-3

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