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Quasitriangular structures on the super-Yangian and quantum loop superalgebra, and difference equations

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Abstract

In the framework of the Toledano Laredo and Gautam approach, we consider the structures of tensor categories on the analogues of the \(\mathfrak{O}\)-category for representations of the super-Yangian \(Y_{\hbar}(A(m,n))\) of the special linear Lie superalgebra and the quantum loop superalgebra \(U_q(LA(m,n))\), and study the relation between them. The basic result is the construction of an isomorphism in the category of Hopf superalgebras between completions of the super-Yangian and quantum loop superalgebra equipped with so-called Drinfeld comultiplications. We formulate the theorem on the equivalence of tensor categories of super-Yangian modules and quantum loop superalgebra modules, which enhances the previous result. We also describe the relation between quasitriangular structures and Abelian difference equations, which are defined by the Abelian parts of the universal \(R\)-matrices.

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References

  1. V. A. Stukopin, “Isomorphism of the Yangian \(Y_{\hbar}(A(m,n))\) of the special linear,” Theoret. and Math. Phys., 198, 129–144 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  2. V. A. Stukopin, “Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra,” Theoret. and Math. Phys., 204, 1227–1243 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  3. V. Stukopin, “On the relationship between super Yangian and quantum loop superalgebra in the case Lie superalgebra \(\mathfrak{sl}(1,1)\),” J. Phys.: Conf. Ser., 1194, 012103, 14 pp. (2019).

    Google Scholar 

  4. S. Gautam and V. Toledano Laredo, “Yangians and quantum loop algebras,” Selecta Math., 19, 271–336 (2013).

    Article  MathSciNet  Google Scholar 

  5. S. Gautam and V. Toledano Laredo, “Yangians, quantum loop algebras and abelian difference equations,” J. Amer. Math. Soc., 29, 775–824 (2016).

    Article  MathSciNet  Google Scholar 

  6. S. Gautam and V. Toledano Laredo, “Meromorphic tensor equivalence for Yangians and quantum loop algebras,” Publ. Math. Inst. Hautes Études Sci., 125, 267–337 (2017).

    Article  MathSciNet  Google Scholar 

  7. S.-J. Kang, M. Kashiwara, and S.-J. Oh, “Supercategorification of quantum Kac–Moody algebras II,” Adv. Math., 265, 169–240 (2014).

    Article  MathSciNet  Google Scholar 

  8. A. Mazurenko and V. A. Stukopin, “Classification of Hopf superalgebras associated with quantum special linear superalgebra at roots of unity using Weyl groupoid,” arXiv: 2111.06576.

  9. V. A. Stukopin, “The Yangian Double of the Lie Superalgebra \(A(m,n)\),” Funct. Anal. Appl., 40, 155–158 (2006).

    Article  MathSciNet  Google Scholar 

  10. V. A. Stukopin, “The quantum double of the Yangian of the Lie superalgebra \(A(m,n)\) and computation of the universal \(R\)-matrix,” J. Math. Sci., 142, 1989–2006 (2007).

    Article  MathSciNet  Google Scholar 

  11. S. M. Khoroshkin and V. N. Tolstoy, “Universal \(R\)-matrix for quantized (super)algebras,” Commun. Math. Phys., 141, 599–617 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  12. S. Levendorskii, Ya. Soibel’man, and V. Stukopin, “Quantum Weyl group and universal \(R\)-matrix for quantum affine Lie algebra \(A^{(1)}_1\),” Lett. Math. Phys., 27, 253–264 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  13. J. Brundan and A. P. Ellis, “Monoidal supercategories,” Commun. Math. Phys., 351, 1045–1089 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  14. H. Zhang, “Representations of quantum affine superalgebras,” Math. Z., 278, 663–703 (2014); arXiv: 1309.5250.

    Article  MathSciNet  Google Scholar 

  15. V. A. Stukopin, “Representations of the Yangian of a Lie superalgebra of type \(A(m,n)\),” Izv. Math., 77, 1021–1043 (2013).

    Article  MathSciNet  Google Scholar 

  16. A. Molev, Yangians and Classical Lie Algebras, Mathematical Surveys and Monographs, Vol. 143, AMS, Providence, RI (2007).

    Book  Google Scholar 

  17. V. Chari and A. Pressley, A Quide to Quantum Groups, Cambridge Univ. Press, Cambridge (1995).

    MATH  Google Scholar 

  18. V. G. Drinfeld, “A new realization of Yangians and of quantum affine algebras,” Dokl. Math., 36, 212–216 (1988).

    MathSciNet  Google Scholar 

  19. V. G. Drinfel’d, “Quantum groups,” in: Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley, CA, August 3–11, 1986, A. M. Gleason, eds.), ICM, Berkley, CA (1988), pp. 789–820.

    Google Scholar 

  20. L. Frappat, A. Sciarrino, and P. Sorba, Dictionary on Lie Algebras and Superalgebras, Academic Press, London (2000).

    MATH  Google Scholar 

  21. V. G. Kac, “A sketch of Lie superalgebra theory,” Commun. Math. Phys., 53, 31–64 (1977).

    Article  ADS  MathSciNet  Google Scholar 

  22. V. A. Stukopin, “On representations of Yangian of Lie superalgebra \(A(n,n)\) type,” J. Phys.: Conf. Ser., 411, 012027, 13 pp. (2013).

    MathSciNet  Google Scholar 

  23. V. G. Drinfel’d, “Hopf algebras and the quantum Yang-Baxter equation,” Sov. Math. Dokl., 32, 256–258 (1985).

    MATH  Google Scholar 

  24. V. G. Drinfeld, “Degenerate affine Hecke algebras and Yangians,” Funct. Anal. Appl., 20, 58–60 (1986).

    Article  MathSciNet  Google Scholar 

  25. A. I. Molev, “Yangians and their applications,” (Handbook of Algebra, Vol. 3), Elsevier, Amsterdam (2003), pp. 907–959; arXiv: math.QA/0211288.

  26. V. A. Stukopin, “Yangians of Lie Superalgebras of Type \(A(m,n)\),” Funct. Anal. Appl., 28, 217–219 (1994).

    Article  MathSciNet  Google Scholar 

  27. M. L. Nazarov, “Quantum Berezinian and the classical Capelly identity,” Lett. Math. Phys., 21, 123–131 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  28. V. A. Stukopin, “The Yangian of the strange Lie superalgebra and its quantum double,” Theoret. and Math. Phys., 174, 122–133 (2013).

    Article  ADS  MathSciNet  Google Scholar 

  29. V. Stukopin, “Yangian of the strange Lie superalgebra of \(Q_{n-1}\) type, Drinfel’d approach,” SIGMA, 3, 069, 12 pp. (2007); arXiv: 0705.3250.

    MathSciNet  MATH  Google Scholar 

  30. V. Stukopin, “Twisted Yangians, Drinfel’d approach,” J. Math. Sci. (N. Y.), 161, 143–162 (2009).

    Article  MathSciNet  Google Scholar 

  31. L. Dolan, Ch. R. Nappi, and E. Witten, “Yangian symmetry in \({D=4}\) superconformal Yang– Mills theory,” in: Quantum Theory and Symmetries (Cincinnati, Ohio, USA, 10–14 September, 2003, P. C. Argyres, L. C. R. Wijewardhana, F. Mansouri, J. J. Scanio, T. J. Hodges, and P. Suranyi, eds.), World Sci., Singapore (2004), pp. 300–315; arXiv: hep-th/0401243.

    Chapter  Google Scholar 

  32. F. Spill and A. Torrielli, “On Drinfeld’s second realization of the AdS/CFT \(\mathfrak{su}(2|2)\) Yangian,” J. Geom. Phys., 59, 489–502 (2009); arXiv: 0803.3194.

    Article  ADS  MathSciNet  Google Scholar 

  33. E. Frenkel and N. Reshetikhin, “The \(q\)-characters of representations of quantum affine algebras and deformations of \(\mathscr{W}\)-algebras,” in: Recent Developments in Quantum Affine Algebras and Related Topics (North Carolina State University, Raleigh, NC, May 21–24, 1998, Contemporary Mathematics, Vol. 248, N. Jing and K. C. Misra, eds.), AMS, Providence, RI (1999), pp. 163–205.

    Chapter  Google Scholar 

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Funding

The paper is supported by the Russian Science Foundation (grant No. 21-11-00283). Some of the presented results were thought over during the author’s visit to IHES (Bures-sur-Ivette, France), where his work was also supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT grant agreement 677368).

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia

    V. A. Stukopin

  2. Southern Mathematical Institute, Vladikavkaz Scientific Center, Russian Academy of Sciences, Vladikavkaz, Russia

    V. A. Stukopin

  3. Moscow Center for Continuous Mathematical Education, Moscow, Russia

    V. A. Stukopin

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  1. V. A. Stukopin
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Correspondence to V. A. Stukopin.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 213, pp. 129–148 https://doi.org/10.4213/tmf10233.

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Stukopin, V.A. Quasitriangular structures on the super-Yangian and quantum loop superalgebra, and difference equations. Theor Math Phys 213, 1423–1440 (2022). https://doi.org/10.1134/S0040577922100099

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