Vertex representations for Yangians of Kac-Moody algebras
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 665-706.

Using vertex operators, we build representations of the Yangian of a simply laced Kac-Moody algebra and of its double. As a corollary, we prove the Poincaré-Birkhoff-Witt property for simply laced affine Yangians.

À l’aide d’opérateurs vertex, nous construisons des représentations du Yangien d’une algèbre de Kac-Moody simplement lacée et de son double. Comme corollaire, nous démontrons la propriété de Poincaré-Birkhoff-Witt pour les Yangiens affines simplement lacés.

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DOI: 10.5802/jep.103
Classification: 17B37,  81R10,  17B69
Keywords: Yangian, vertex operator, Kac-Moody algebra, Fock space, twisted group algebra, central extension
Guay, Nicolas 1; Regelskis, Vidas 2, 3; Wendlandt, Curtis 1

1 University of Alberta, Department of Mathematical and Statistical Sciences Edmonton, AB T6G 2G1, Canada
2 University of York, Department of Mathematics York, YO10 5DD, UK
3 Vilnius University, Institute of Theoretical Physics and Astronomy Saulėtekio av. 3, Vilnius 10257, Lithuania
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Guay, Nicolas; Regelskis, Vidas; Wendlandt, Curtis. Vertex representations for Yangians of Kac-Moody algebras. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 665-706. doi : 10.5802/jep.103. http://www.numdam.org/articles/10.5802/jep.103/

[AG19] Appel, A.; Gautam, S. An explicit isomorphism between quantum and classical 𝔰𝔩 n , Transform. Groups (2019), 36 pages | arXiv | DOI

[AMR06] Arnaudon, Daniel; Molev, Alexander; Ragoucy, Eric On the R-matrix realization of Yangians and their representations, Ann. Henri Poincaré, Volume 7 (2006) no. 7-8, pp. 1269-1325 | DOI | MR | Zbl

[Ber89] Bernard, Denis Vertex operator representations of the quantum affine algebra 𝒰 q (B r (1) ), Lett. Math. Phys., Volume 17 (1989) no. 3, pp. 239-245 | DOI | MR

[BT19] Bershtein, Mikhail; Tsymbaliuk, Alexander Homomorphisms between different quantum toroidal and affine Yangian algebras, J. Pure Appl. Algebra, Volume 223 (2019) no. 2, pp. 867-899 | DOI | MR | Zbl

[BTM87] Bernard, Denis; Thierry-Mieg, Jean Level one representations of the simple affine Kac-Moody algebras in their homogeneous gradations, Comm. Math. Phys., Volume 111 (1987) no. 2, pp. 181-246 http://projecteuclid.org/euclid.cmp/1104159538 | DOI | MR | Zbl

[CJ01] Chari, Vyjayanthi; Jing, Naihuan Realization of level one representations of U q (𝔤 ^) at a root of unity, Duke Math. J., Volume 108 (2001) no. 1, pp. 183-197 | DOI | MR | Zbl

[DK00] Ding, J.; Khoroshkin, S. Weyl group extension of quantized current algebras, Transform. Groups, Volume 5 (2000) no. 1, pp. 35-59 | DOI | MR | Zbl

[Dri86] Drinfel’d, V. G. Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen., Volume 20 (1986) no. 1, pp. 69-70 | MR | Zbl

[FJ88] Frenkel, Igor B.; Jing, Nai Huan Vertex representations of quantum affine algebras, Proc. Nat. Acad. Sci. U.S.A., Volume 85 (1988) no. 24, pp. 9373-9377 | DOI | MR | Zbl

[FK81] Frenkel, I. B.; Kac, V. G. Basic representations of affine Lie algebras and dual resonance models, Invent. Math., Volume 62 (1980/81) no. 1, pp. 23-66 | DOI | MR | Zbl

[FLM88] Frenkel, Igor; Lepowsky, James; Meurman, Arne Vertex operator algebras and the Monster, Pure and Applied Mathematics, 134, Academic Press, Inc., Boston, MA, 1988 | MR | Zbl

[FT19] Finkelberg, M.; Tsymbaliuk, A. Shifted quantum affine algebras: integral forms in type A, Arnold Math. J. (2019) | arXiv | DOI

[GNOS86] Goddard, P.; Nahm, W.; Olive, D.; Schwimmer, A. Vertex operators for non-simply-laced algebras, Comm. Math. Phys., Volume 107 (1986) no. 2, pp. 179-212 http://projecteuclid.org/euclid.cmp/1104116020 | DOI | MR | Zbl

[GNW18] Guay, Nicolas; Nakajima, Hiraku; Wendlandt, Curtis Coproduct for Yangians of affine Kac-Moody algebras, Adv. Math., Volume 338 (2018), pp. 865-911 | DOI | MR | Zbl

[GRW19] Guay, Nicolas; Regelskis, Vidas; Wendlandt, Curtis Equivalences between three presentations of orthogonal and symplectic Yangians, Lett. Math. Phys., Volume 109 (2019) no. 2, pp. 327-379 | DOI | MR | Zbl

[GTL13] Gautam, Sachin; Toledano Laredo, Valerio Yangians and quantum loop algebras, Selecta Math. (N.S.), Volume 19 (2013) no. 2, pp. 271-336 | DOI | MR | Zbl

[GTL16] Gautam, Sachin; Toledano Laredo, Valerio Yangians, quantum loop algebras, and abelian difference equations, J. Amer. Math. Soc., Volume 29 (2016) no. 3, pp. 775-824 | DOI | MR | Zbl

[Gua07] Guay, Nicolas Affine Yangians and deformed double current algebras in type A, Adv. Math., Volume 211 (2007) no. 2, pp. 436-484 | DOI | MR | Zbl

[IK96] Iohara, Kenji; Kohno, Mika A central extension of 𝒟Y (𝔤𝔩 2 ) and its vertex representations, Lett. Math. Phys., Volume 37 (1996) no. 3, pp. 319-328 | DOI | Zbl

[Ioh96] Iohara, Kenji Bosonic representations of Yangian double 𝒟Y (𝔤) with 𝔤=𝔤𝔩 N ,𝔰𝔩 N , J. Phys. A, Volume 29 (1996) no. 15, pp. 4593-4621 | DOI | MR

[Jin90] Jing, Nai Huan Twisted vertex representations of quantum affine algebras, Invent. Math., Volume 102 (1990) no. 3, pp. 663-690 | DOI | MR

[Jin98] Jing, Naihuan Quantum Kac-Moody algebras and vertex representations, Lett. Math. Phys., Volume 44 (1998) no. 4, pp. 261-271 | DOI | MR | Zbl

[Jin99] Jing, Naihuan Level one representations of U q (G 2 (1) ), Proc. Amer. Math. Soc., Volume 127 (1999) no. 1, pp. 21-27 | DOI | MR | Zbl

[Jin00] Jing, Naihuan Quantum Z-algebras and representations of quantum affine algebras, Comm. Algebra, Volume 28 (2000) no. 2, pp. 829-844 | DOI | MR | Zbl

[JKM99] Jing, Naihuan; Koyama, Yoshitaka; Misra, Kailash C. Level one representations of quantum affine algebras U q (C n (1) ), Selecta Math. (N.S.), Volume 5 (1999) no. 2, pp. 243-255 | DOI | MR | Zbl

[JM96] Jing, Naihuan; Misra, Kailash C. Vertex operators of level-one U q (B n (1) )-modules, Lett. Math. Phys., Volume 36 (1996) no. 2, pp. 127-143 | DOI | MR | Zbl

[Kac90] Kac, Victor G. Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990 | DOI | Zbl

[Kas84] Kassel, Christian Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra, J. Pure Appl. Algebra, Volume 34 (1984) no. 2-3, pp. 265-275 | DOI | Zbl

[Kho97] Khoroshkin, Sergej M. Central extension of the Yangian double, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995) (Sémin. Congr.), Volume 2, Société Mathématique de France, Paris, 1997, pp. 119-135 | MR | Zbl

[Kod19] Kodera, Ryosuke Affine Yangian action on the Fock space, Publ. RIMS, Kyoto Univ., Volume 55 (2019) no. 1, pp. 189-234 | DOI | MR | Zbl

[KSU97] Kimura, Kazuhiro; Shiraishi, Jun’ichi; Uchiyama, Jun A level-one representation of the quantum affine superalgebra U q (𝔰𝔩 ^(M+1|N+1)), Comm. Math. Phys., Volume 188 (1997) no. 2, pp. 367-378 | DOI | MR | Zbl

[KT96] Khoroshkin, S. M.; Tolstoy, V. N. Yangian double, Lett. Math. Phys., Volume 36 (1996) no. 4, pp. 373-402 | DOI | MR | Zbl

[Lev93] Levendorskiĭ, Serge Z. On PBW bases for Yangians, Lett. Math. Phys., Volume 27 (1993) no. 1, pp. 37-42 | DOI | MR | Zbl

[LL04] Lepowsky, James; Li, Haisheng Introduction to vertex operator algebras and their representations, Progress in Math., 227, Birkhäuser Boston, Inc., Boston, MA, 2004 | DOI | MR | Zbl

[Mol07] Molev, Alexander Yangians and classical Lie algebras, Math.Surveys and Monographs, 143, American Mathematical Society, Providence, RI, 2007 | DOI | MR | Zbl

[MRY90] Moody, Robert V.; Rao, Senapathi Eswara; Yokonuma, Takeo Toroidal Lie algebras and vertex representations, Geom. Dedicata, Volume 35 (1990) no. 1-3, pp. 283-307 | DOI | MR | Zbl

[Nak01] Nakajima, Hiraku Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc., Volume 14 (2001) no. 1, pp. 145-238 | DOI | MR | Zbl

[Neh03] Neher, Erhard An introduction to universal central extensions of Lie superalgebras, Groups, rings, Lie and Hopf algebras (St. John’s, NF, 2001) (Math. Appl.), Volume 555, Kluwer Acad. Publ., Dordrecht, 2003, pp. 141-166 | DOI | MR | Zbl

[Sai98] Saito, Yoshihisa Quantum toroidal algebras and their vertex representations, Publ. RIMS, Kyoto Univ., Volume 34 (1998) no. 2, pp. 155-177 | DOI | MR | Zbl

[YZ18a] Yang, Yaping; Zhao, Gufang Cohomological Hall algebras and affine quantum groups, Selecta Math. (N.S.), Volume 24 (2018) no. 2, pp. 1093-1119 | DOI | MR | Zbl

[YZ18b] Yang, Yaping; Zhao, Gufang The PBW theorem for the affine Yangians, 2018 | arXiv

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