Geodesic
$R$-Matrix Presentation of Quantum Affine Superalgebra for Type $\mathfrak{osp}(2m+1|2n)$
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In our preceding research, we introduced the Drinfeld presentation of the quantum affine superalgebra associated to the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for $m>0$. We provided the isomorphism between its Drinfeld–Jimbo presentation and Drinfeld presentation using braid group actions as a fundamental method. Based on this work, our current study delves into its $R$-matrix presentation, wherein we establish a clear isomorphism between the $R$-matrix presentation and the Drinfeld presentation. In particular, our contribution extends the investigations of Jing, Liu and Molev concerning quantum affine algebra in type BCD to the realm of supersymmetry.
Keywords: quantum affine superalgebra, $R$-matrix presentation, Drinfeld presentation
Mots-clés : universal $R$-matrix. 
@article{SIGMA_2024_20_a104,
     author = {Xianghua Wu and Hongda Lin and Honglian Zhang},
     title = {$R${-Matrix} {Presentation} of {Quantum} {Affine} {Superalgebra} for {Type} $\mathfrak{osp}(2m+1|2n)$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
     volume = {20},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a104/}
}
TY  - JOUR
AU  - Xianghua Wu
AU  - Hongda Lin
AU  - Honglian Zhang
TI  - $R$-Matrix Presentation of Quantum Affine Superalgebra for Type $\mathfrak{osp}(2m+1|2n)$
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2024
VL  - 20
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a104/
LA  - en
ID  - SIGMA_2024_20_a104
ER  - 
%0 Journal Article
%A Xianghua Wu
%A Hongda Lin
%A Honglian Zhang
%T $R$-Matrix Presentation of Quantum Affine Superalgebra for Type $\mathfrak{osp}(2m+1|2n)$
%J Symmetry, integrability and geometry: methods and applications
%D 2024
%V 20
%U http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a104/
%G en
%F SIGMA_2024_20_a104
Xianghua Wu; Hongda Lin; Honglian Zhang. $R$-Matrix Presentation of Quantum Affine Superalgebra for Type $\mathfrak{osp}(2m+1|2n)$. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a104/

[1] Beck J., “Braid group action and quantum affine algebras”, Comm. Math. Phys., 165 (1994), 555–568, arXiv: hep-th/9404165 | DOI | MR | Zbl

[2] Bezerra L., Futorny V., Kashuba I., Drinfeld realization for quantum affine superalgebras of type $B$, arXiv: 2405.05533

[3] Brundan J., Kleshchev A., “Parabolic presentations of the Yangian $Y({\mathfrak{gl}}_n)$”, Comm. Math. Phys., 254 (2005), 191–220, arXiv: math.QA/0407011 | DOI | MR | Zbl

[4] Cai J.-F., Wang S.-K., Wu K., Zhao W.-Z., “Drinfeld realization of quantum affine superalgebra $U_q(\widehat{\mathfrak{gl}(1|1)})$”, J. Phys. A, 31 (1998), 1989–1994, arXiv: q-alg/9703022 | DOI | MR | Zbl

[5] Damiani I., “Drinfeld realization of affine quantum algebras: the relations”, Publ. Res. Inst. Math. Sci., 48 (2012), 661–733, arXiv: 1406.6729 | DOI | MR | Zbl

[6] Damiani I., “From the Drinfeld realization to the Drinfeld–Jimbo presentation of affine quantum algebras: injectivity”, Publ. Res. Inst. Math. Sci., 51 (2015), 131–171, arXiv: 1407.0341 | DOI | MR | Zbl

[7] Ding J.T., Frenkel I.B., “Isomorphism of two realizations of quantum affine algebra $U_q(\mathfrak{gl}(n))$”, Comm. Math. Phys., 156 (1993), 277–300 | DOI | MR | Zbl

[8] Drinfeld V.G., “A new realization of Yangians and of quantum affine algebras”, Soviet Math. Dokl., 36 (1988), 212–216 | MR | Zbl

[9] Drinfeld V.G., “Hopf algebras and the quantum Yang–Baxter equation”, Yang–Baxter Equation in Integrable Systems, Adv. Ser. Math. Phys., World Scientific Publishing, 1990, 264–268 | DOI

[10] Fan H., Hou B.-Y., Shi K.-J., “Drinfeld constructions of the quantum affine superalgebra $U_q(\widehat {\mathfrak{gl}(m|n)})$”, J. Math. Phys., 38 (1997), 411–433 | DOI | MR | Zbl

[11] Frassek R., Tsymbaliuk A., Orthosymplectic Yangians, arXiv: 2311.18818

[12] Frenkel E., Mukhin E., “The Hopf algebra $\operatorname{Rep} U_q\widehat{\mathfrak{gl}}_\infty$”, Selecta Math. (N.S.), 8 (2002), 537–635, arXiv: math.QA/0103126 | DOI | MR | Zbl

[13] Frenkel I.B., Reshetikhin N.Yu., “Quantum affine algebras and holonomic difference equations”, Comm. Math. Phys., 146 (1992), 1–60 | DOI | MR | Zbl

[14] Galleas W., Martins M.J., “New $R$-matrices from representations of braid-monoid algebras based on superalgebras”, Nuclear Phys. B, 732 (2006), 444–462 | DOI | MR | Zbl

[15] Gelfand I.M., Retakh V.S., “Determinants of matrices over noncommutative rings”, Funct. Anal. Appl., 25 (1991), 91–102 | DOI | MR | Zbl

[16] Gould M.D., Zhang Y.-Z., “On super-RS algebra and Drinfeld realization of quantum affine superalgebras”, Lett. Math. Phys., 44 (1998), 291–308, arXiv: q-alg/9712011 | DOI | MR | Zbl

[17] Jimbo M., “A $q$-difference analogue of $U(\mathfrak{g})$ and the Yang–Baxter equation”, Lett. Math. Phys., 10 (1985), 63–69 | DOI | MR | Zbl

[18] Jimbo M., “Quantum $R$ matrix for the generalized Toda system”, Comm. Math. Phys., 102 (1986), 537–547 | DOI | MR | Zbl

[19] Jing N., Liu M., Molev A., “Isomorphism between the $R$-matrix and Drinfeld presentations of Yangian in types $B$, $C$ and $D$”, Comm. Math. Phys., 361 (2018), 827–872, arXiv: 1705.08155 | DOI | MR | Zbl

[20] Jing N., Liu M., Molev A., “Isomorphism between the $R$-matrix and Drinfeld presentations of quantum affine algebra: type $C$”, J. Math. Phys., 61 (2020), 031701, 41 pp., arXiv: 1903.00204 | DOI | MR | Zbl

[21] Jing N., Liu M., Molev A., “Isomorphism between the $R$-matrix and Drinfeld presentations of quantum affine algebra: types $B$ and $D$”, SIGMA, 16 (2020), 043, 49 pp., arXiv: 1911.03496 | DOI | MR | Zbl

[22] Jing N., Zhang H., “Drinfeld realization of twisted quantum affine algebras”, Comm. Algebra, 35 (2007), 3683–3698 | DOI | MR | Zbl

[23] Jing N., Zhang H., “Drinfeld realization of quantum twisted affine algebras via braid group”, Adv. Math. Phys., 2016 (2016), 4843075, 15 pp., arXiv: 1301.3550 | DOI | MR | Zbl

[24] Kac V.G., “Lie superalgebras”, Adv. Math., 26 (1977), 8–96 | DOI | MR | Zbl

[25] Khoroshkin S.M., Tolstoy V.N., Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan–Weyl realizations for quantum affine algebras, arXiv: hep-th/9404036

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

[27] Krob D., Leclerc B., “Minor identities for quasi-determinants and quantum determinants”, Comm. Math. Phys., 169 (1995), 1–23, arXiv: hep-th/9411194 | DOI | MR | Zbl

[28] Levendorskiĭ S.Z., “On generators and defining relations of Yangians”, J. Geom. Phys., 12 (1993), 1–11 | DOI | MR | Zbl

[29] Lin H., Yamane H., Zhang H., “On generators and defining relations of quantum affine superalgebra $U_q (\widehat{\mathfrak{sl}}_{m|n})$”, J. Algebra Appl., 23 (2024), 2450021, 29 pp. | DOI | MR | Zbl

[30] Lu K., “Isomorphism between twisted $q$-Yangians and affine $i$quantum groups: type AI”, Int. Math. Res. Not. (to appear) , arXiv: 2308.12484 | DOI | MR

[31] Mehta M., Dancer K.A., Gould M.D., Links J., “Generalized Perk–Schultz models: solutions of the Yang–Baxter equation associated with quantized orthosymplectic superalgebras”, J. Phys. A, 39 (2006), L17–L26, arXiv: nlin.SI/0509019 | DOI | MR | Zbl

[32] Molev A.I., “A Drinfeld-type presentation of the orthosymplectic Yangians”, Algebr. Represent. Theory, 27 (2024), 469–494, arXiv: 2112.10419 | DOI | MR | Zbl

[33] Reshetikhin N.Yu., Semenov-Tian-Shansky M.A., “Central extensions of quantum current groups”, Lett. Math. Phys., 19 (1990), 133–142 | DOI | MR | Zbl

[34] van de Leur J.W., “A classification of contragredient Lie superalgebras of finite growth”, Comm. Algebra, 17 (1989), 1815–1841 | DOI | MR | Zbl

[35] Wu X., Lin H., Zhang H., Braid group action and quantum affine superalgebra for type $\mathfrak{osp}(2m+1|2n)$, arXiv: 2410.21755

[36] Xu Y., Zhang R., Drinfeld realisations and vertex operator respresentations of the quantum affine superalgebras, arXiv: 1802.09702

[37] Yamane H., “On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras”, Publ. Res. Inst. Math. Sci., 35 (1999), 321–390, arXiv: q-alg/9603015 | DOI | MR | Zbl

[38] Zhang H., “RTT realization of quantum affine superalgebras and tensor products”, Int. Math. Res. Not., 2016 (2016), 1126–1157, arXiv: 1407.7001 | DOI | MR | Zbl

[39] Zhang H., Jing N., “Drinfeld realization of twisted quantum affine algebra”, Comm. Algebra, 35 (2007), 3683–3698, arXiv: 1301.3550 | DOI | MR | Zbl

[40] Zhang Y.-Z., “Comments on the {D}rinfeld realization of the quantum affine superalgebra $U_q[\mathfrak{gl}(m|n)^{(1)}]$ and its {H}opf algebra structure”, J. Phys. A, 30 (1997), 8325–8335, arXiv: q-alg/9703020 | DOI | MR | Zbl