Geodesic
Quasi-triangular structures on the super-Yangian and quantum loop superalgebra and difference equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 213 (2022) no. 1, pp. 129-148 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Following the V. Toledano-Laredo and S. Gautam approach we consider structures of tensor categories on analogues of the category $\mathfrak{O} $ for representations of the super Yangian $Y_ {\ hbar} (A (m, n)) $ of the special linear superalgebra Lie and the quantum loop superalgebra $U_q (LA (m, n)) $, we investigate the connection between them. The connection between Quasi-triangular structures and Abelian difference equations, which are determined by the Abelian parts of the universal R-matrices, is also described. Bibliography: 34 titles.
Keywords: Yangian of Lie superalgebra, quantum loop superalgebra, Yangian module, category of $\mathfrak{O}$ representations, Lie superalgebra, Hopf superalgebra, tensor category, difference equations.
Mots-clés : universal R-matrix, quasitriangular structure 
@article{TMF_2022_213_1_a7,
     author = {V. A. Stukopin},
     title = {Quasi-triangular structures on the {super-Yangian} and quantum loop superalgebra and difference equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {129--148},
     year = {2022},
     volume = {213},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_213_1_a7/}
}
TY  - JOUR
AU  - V. A. Stukopin
TI  - Quasi-triangular structures on the super-Yangian and quantum loop superalgebra and difference equations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2022
SP  - 129
EP  - 148
VL  - 213
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2022_213_1_a7/
LA  - ru
ID  - TMF_2022_213_1_a7
ER  - 
%0 Journal Article
%A V. A. Stukopin
%T Quasi-triangular structures on the super-Yangian and quantum loop superalgebra and difference equations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2022
%P 129-148
%V 213
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2022_213_1_a7/
%G ru
%F TMF_2022_213_1_a7
V. A. Stukopin. Quasi-triangular structures on the super-Yangian and quantum loop superalgebra and difference equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 213 (2022) no. 1, pp. 129-148. http://geodesic.mathdoc.fr/item/TMF_2022_213_1_a7/

[1] V. A. Stukopin, “Ob izomorfizme yangiana $Y_\hbar(A(m,n))$ spetsialnoi lineinoi superalgebry Li i kvantovoi petelevoi superalgebry $U_q(LA(m,n))$”, TMF, 198:1 (2019), 145–161 | DOI | DOI | MR

[2] V. A. Stukopin, “O svyazi kategorii predstavlenii yangiana spetsialnoi lineinoi superalgebry Li i kvantovoi petlevoi superalgebry”, TMF, 204:3 (2020), 466–484 | DOI | DOI | MR

[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 (2019), 012103, 14 pp. | DOI

[4] S. Gautam, V. Toledano Laredo, “Yangians and quantum loop algebras”, Selecta Math., 19:3 (2013), 271–336 | DOI | MR

[5] S. Gautam, V. Toledano Laredo, “Yangians, quantum loop algebras and abelian difference equations”, J. Amer. Math. Soc., 29:3 (2016), 775–824 | DOI | MR | Zbl

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

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

[8] A. Mazurenko, 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, “O duble yangiana superalgebry Li tipa $A(m,n)$”, Funkts. analiz i ego pril., 40:2 (2006), 81–84 | DOI | DOI | MR | Zbl

[10] V. A. Stukopin, “Kvantovyi dubl yangiana superalgebry Li tipa $A(m,n)$ i vychislenie universalnoi $R$-matritsy”, Fundament. i prikl. matem., 11:2 (2005), 185–208 | DOI | MR | Zbl

[11] S. M. Khoroshkin, V. N. Tolstoy, “Universal $R$-matrix for quantized (super)algebras”, Commun. Math. Phys., 141:3 (1991), 599–617 | DOI | MR

[12] S. Levendorskii, Ya. Soibel'man, V. Stukopin, “Quantum Weyl group and universal $R$-matrix for quantum affine Lie algebra $A^{(1)}_1$”, Lett. Math. Phys., 27:4 (1993), 253–264 | DOI | MR

[13] J. Brundan, A. P. Ellis, “Monoidal supercategories”, Commun. Math. Phys., 351:3 (2017), 1045–1089 | DOI | MR

[14] H. Zhang, “Representations of quantum affine superalgebras”, Math. Z., 278:3–4 (2014), 663–703, arXiv: 1309.5250 | DOI | MR

[15] V. A. Stukopin, “O predstavleniyakh yangiana superalgebry Li tipa $A(m,n)$”, Izv. RAN. Ser. matem., 77:5 (2013), 179–202 | DOI | DOI | MR | Zbl

[16] A. I. Molev, Yangiany i klassicheskie algebry Li, MTsNMO, M., 2009

[17] V. Chari, A. Pressley, A Quide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1995 | MR

[18] V. G. Drinfeld, “Novaya realizatsiya yangianov i kvantovannykh affinnykh algebr”, Dokl. AN SSSR, 296:1 (1987), 13–17 | MR | Zbl

[19] V. G. Drinfel'd, “Quantum groups”, Proceedings of the International Congress of Mathematicians (Berkeley, CA, August 3–11, 1986), v. 1, ed. A. M. Gleason, ICM, Berkley, CA, 1988, 789–820 | MR | Zbl

[20] L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Algebras and Superalgebras, Academic Press, London, 2000 | MR

[21] V. G. Kac, “A sketch of Lie superalgebra theory”, Commun. Math. Phys., 53:1 (1977), 31–64 | DOI | MR

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

[23] V. G. Drinfeld, “Algebry Khopfa i kvantovoe uravnenie Yanga–Bakstera”, Dokl. AN SSSR, 283:5 (1985), 1060–1064 | MR | Zbl

[24] V. G. Drinfeld, “Vyrozhdennye affinnye algebry Gekke i yangiany”, Funkts. analiz i ego pril., 20:1 (1986), 69–70 | DOI | MR | Zbl

[25] A. I. Molev, “Yangians and their applications”, Handbook of Algebra, 3, Elsevier, Amsterdam, 2003, 907–959, arXiv: math.QA/0211288 | DOI | MR

[26] V. A. Stukopin, “O yangianakh superalgebr Li tipa $A(m,n)$”, Funkts. analiz i ego pril., 28:3 (1994), 85–88 | DOI | MR | Zbl

[27] M. L. Nazarov, “Quantum Berezinian and the classical Capelly identity”, Lett. Math. Phys., 21:2 (1991), 123–131 | DOI | MR

[28] V. A. Stukopin, “Yangian strannoi superalgebry Li i ego kvantovyi dubl”, TMF, 174:1 (2013), 140–153 | DOI | DOI | MR | Zbl

[29] V. Stukopin, “Yangian of the strange Lie superalgebra of $Q_{n-1}$ type, Drinfel'd approach”, SIGMA, 3 (2007), 069, 12 pp., arXiv: 0705.3250 | DOI | MR

[30] V. Stukopin, “Twisted Yangians, Drinfel'd approach”, J. Math. Sci. (N. Y.), 161:1 (2009), 143–162 | DOI | MR

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

[32] F. Spill, A. Torrielli, “On Drinfeld's second realization of the AdS/CFT $\mathfrak{su}(2|2)$ Yangian”, J. Geom. Phys., 59:4 (2009), 489–502, arXiv: 0803.3194 | DOI | MR

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