The Weyl groupoid and its action on the affine super-Yangian

V. D. Volkov   ; V. A. Stukopin

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 30, Tome 532 (2024), pp. 119-135

Résumé

We define the action of the Weyl groupoid on the affine super-Yangian $Y_{\hbar}(\widehat{sl}(m|n, \Pi))$ of the special linear Kac-Moody superalgebra $\hat{sl}(m|n, \Pi) $, given by an arbitrary system of simple roots $\Pi$. Affine super-Yangians of this type form a category. Morphisms in this category are given by the action of the elements of the Weyl groupoid. All super-Yangians from this category are isomorphic as associative superalgebras, but morphisms defined by the action of elements of a Weyl groupoid do not preserve coproducts. We describe coproducts on super-Yangians and their relation to the Weyl groupoid action.

Bibliographie

[1] V. Chari, A. Pressley, A guide to quantum groups, Camb.Univ.Press, Cambridge, 1995 | MR | Zbl

[2] V. G. Drinfeld, “Novaya realizatsiya yangianov i kvantovykh affinnykh algebr”, Doklady AN SSSR, 36 (1988), 212–216 | Zbl

[3] S. I. Boyarchenko, S. Z. Levendorskii, “On affine Yangians”, Lett. Math. Phys., 32:4 (1993), 2691–274 | MR

[4] N. Guay, “Affine Yangians and deformed double current algebras in type A”, Adv. Math., 211 (2007), 426–484 | DOI | MR

[5] N. Guay, H. Nakajima, C. Wendlandt, “Coproduct for Yangians of affine Kac-Moody algebras on generators and defining relations of Yangians”, Adv. Math., 338 (2018), 865–911 | DOI | MR | Zbl

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

[7] V. Drinfeld, “Quantum groups”, Proc. Int. Cong. Math., v. 1, Berkley, 1988, 789–820 | MR

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

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

[10] V. A. Stukopin, “O svyazi kategorii predstavlenii yangiana spetsialnoi lineinoi superalgebry Li i kvantovoi petelevoi superalgebry”, Teor. i matem. fizika, 204:3 (2020), 466–484 | DOI | MR | Zbl

[11] V. D. Volkov, V. A. Stukopin, “Affinnyi superyangian i kvantovyi gruppoid Veilya”, Teor. i matem. fizika, 216:3 (2023), 476–489 | DOI | MR | Zbl

[12] D. Galakhov, M. Yamazaki, “Quiver Yangian and supersymmetric quantum mechanics”, Commun. Math. Phys., 396:2 (2022), 713–785 | DOI | MR | Zbl

[13] D. Galakhov, A. Morozov, N. Tselousov, Quiver Yangian and supersymmetric quantum mechanics, arXiv: 2402.05920v2 [hep-th]

[14] D. Maulik, A. Okounkov, Quantum groups and quantum cohomology, arXiv: 1211.1287v2 [math.AG] | MR

[15] A. Mazurenko, V. Stukopin, Classification of Hopf superalgebras associated with quantum special linear superalgebra at roots of unity using Weyl groupoid, arXiv: 2111.06576 [math.QA]

[16] V. A. Stukopin, “O duble yangiana superalgebry Li tipa $A(m,n)$”, Funkts. analiz i ego prilozh., 40:2 (2006), 81–84 | MR

[17] V. Stukopin, “Yangians of classical Lie superalgebras: basic constructions, quantum double and universal $R$-matrix”, Proceedings of Institute of Mathematics of NAS of Ukraine, 50:3 (2004), 1195–1201 | MR | Zbl

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

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

[20] I. M. Musson, Lie superalgebras and enveloping algebras, Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 2012 | DOI | MR | Zbl

[21] C. Peng M. R. Gaberdiel, W. Li, H. Zhang, The supersymmetric affine Yangian, 2018, arXiv: 1711.07449v.2 [math.hep-th] | MR

[22] V. A. Stukopin, A. Mazurenko, Classification of Hopf superalgebra structures on Drinfeld super Yangians, 2022, arXiv: 2210.08365 [math.QA] | MR | Zbl

[23] Ryosuke Kodera, “Braid group action on affine Yangian”, Symmetry, Integrability and Geometry: Methods and Applications, 15 (2019), 28 | MR | Zbl

[24] Mamoru Ueda, Construction of Affine Super Yangian, 2021, arXiv: 1911.0666v6 [math. RT] | MR