Geodesic
Current presentation for the super-Yangian double $DY(\mathfrak{gl}(m|n))$ and Bethe vectors
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 1, pp. 33-99 Cet article a éte moissonné depuis la source Math-Net.Ru

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Bethe vectors are found for quantum integrable models associated with the supersymmetric Yangians $Y(\mathfrak{gl}(m|n)$ in terms of the current generators of the Yangian double $DY(\mathfrak{gl}(m|n))$. The method of projections onto intersections of different types of Borel subalgebras of this infinite-dimensional algebra is used to construct the Bethe vectors. Calculation of these projections makes it possible to express the supersymmetric Bethe vectors in terms of the matrix elements of the universal monodromy matrix. Two different presentations for the Bethe vectors are obtained by using two different but isomorphic current realizations of the Yangian double $DY(\mathfrak{gl}(m|n))$. These Bethe vectors are also shown to obey certain recursion relations which prove their equivalence. Bibliography: 30 titles.
Keywords: Bethe vector, current algebra, projection.
Mots-clés : monodromy matrix, Gauss decomposition 
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A. A. Hutsalyuk; A. Liashyk; S. Z. Pakulyak; E. Ragoucy; N. A. Slavnov. Current presentation for the super-Yangian double $DY(\mathfrak{gl}(m|n))$ and Bethe vectors. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 1, pp. 33-99. http://geodesic.mathdoc.fr/item/RM_2017_72_1_a1/

[1] L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi i $XYZ$ model Geizenberga”, UMN, 34:5(209) (1979), 13–63 | MR

[2] E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, “Kvantovyi metod obratnoi zadachi. I”, TMF, 40:2 (1980), 194–220 ; E. K. Sklyanin, L. A. Takhtadzhyan, L. D. Faddeev, “Quantum inverse problem method. I”, Theoret. and Math. Phys., 40:2 (1979), 688–706 | MR | Zbl | DOI

[3] N. A. Slavnov, “Vychislenie skalyarnykh proizvedenii volnovykh funktsii i formfaktorov v ramkakh algebraicheskogo anzatsa Bete”, TMF, 79:2 (1989), 232–240 | MR

[4] V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Monogr. Math. Phys., Cambridge Univ. Press, Cambridge, 1993, xx+555 pp. | DOI | MR | Zbl

[5] N. A. Slavnov, “Algebraicheskii anzats Bete i kvantovye integriruemye sistemy”, UMN, 62:4(376) (2007), 91–132 | DOI | MR | Zbl

[6] P. P. Kulish, N. Yu. Reshetikhin, “Diagonalisation of $\operatorname{GL}(N)$ invariant transfer matrices and quantum $N$-wave system (Lee model)”, J. Phys. A, 16 (1983), L591–L596 | DOI | MR | Zbl

[7] P. P. Kulish, N. Yu. Reshetikhin, “Obobschennyi ferromagnetik Geizenberga i model Grossa–Neve”, ZhETF, 80:1 (1981), 214–228 | MR

[8] P. P. Kulish, N. Yu. Reshetikhin, “O $GL_3$-invariantnykh resheniyakh uravneniya Yanga–Bakstera i assotsiirovannykh kvantovykh sistemakh”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 3, Zap. nauch. sem. LOMI, 120, Izd-vo “Nauka”, Leningrad. otd., L., 1982, 92–121 ; P. P. Kulish, N. Yu. Reshetikhin, “$GL_3$ invariant solutions of the Yang–Baxter equation and associated quantum systems”, J. Soviet Math., 34:5 (1986), 1948–1971 | MR

[9] N. Yu. Reshetikhin, “Vychislenie normy betevskikh vektorov v modelyakh s $SU(3)$ simmetriei”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 6, Zap. nauch. sem. LOMI, 150, Izd-vo “Nauka”, Leningrad. otd., L., 1986, 196–213 ; N. Yu. Reshetikhin, “Calculation of the norm of Bethe vectors in models with $SU(3)$-symmetry”, J. Soviet Math., 46:1 (1989), 1694–1706 | MR | DOI

[10] A. N. Varchenko, V. O. Tarasov, “Dzheksonovskie integralnye predstavleniya dlya reshenii kvantovogo uravneniya Knizhnika–Zamolodchikova”, Algebra i analiz, 6:2 (1994), 90–137 ; A. Varchenko, V. Tarasov, “Jackson integral representations for solutions to the quantized Knizhnik–Zamolodchikov equation”, St. Petersburg Math. J., 6:2 (1995), 275–313; (1994 (v1 – 1993)), 50 pp., arXiv: hep-th/9311040 | MR | Zbl

[11] V. Tarasov, A. Varchenko, “Combinatorial formulae for nested Bethe vectors”, SIGMA, 9 (2013), 048, 28 pp. ; (2013 (v1 – 2007)), 28 pp., arXiv: math/0702277 | DOI | MR | Zbl

[12] S. Belliard, E. Ragoucy, “The nested Bethe ansatz for ‘all’ closed spin chains”, J. Phys. A, 41:29 (2008), 295202, 33 pp. | DOI | MR | Zbl

[13] S. Khoroshkin, S. Pakuliak, “A computation of an universal weight function for quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$”, J. Math. Kyoto Univ., 48:2 (2008), 277–321 ; (2007), 40 pp., arXiv: 0711.2819 | MR | Zbl

[14] A. F. Oskin, S. Z. Pakulyak, A. V. Silantev, “Ob universalnoi vesovoi funktsii dlya kvantovoi affinnoi algebry $U_q(\widehat{\mathfrak{gl}}_N)$”, Algebra i analiz, 21:4 (2009), 196–240 ; A. Os'kin, S. Pakuliak, A. Silant'ev, “On the universal weight function for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$”, St. Petersburg Math. J., 21:4 (2010), 651–680 ; (2007), 35 pp., arXiv: 0711.2821 | MR | Zbl | DOI

[15] S. Belliard, S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Bethe vectors of $GL(3)$-invariant integrable models”, J. Stat. Mech. Theory Exp., 2013, no. 2, P02020, 24 pp. ; 2013 (v1 – 2012), 22 pp., arXiv: 1210.0768 | MR

[16] S. Belliard, S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Form factors in $SU(3)$-invariant integrable models”, J. Stat. Mech. Theory Exp., 2013, no. 4, P04033, 16 pp. ; 2013 (v1 – 2012), 15 pp., arXiv: 1211.3968 | MR

[17] S. Belliard, S. Pakuliak, E. Ragoucy, N. A. Slavnov, “Bethe vectors of quantum integrable models with $\operatorname{GL}(3)$ trigonometric $R$-matrix”, SIGMA, 9 (2013), 058, 23 pp. | DOI | MR | Zbl

[18] A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, N. A. Slavnov, “Form factors of the monodromy matrix entries in $\mathfrak{gl}(2|1)$-invariant integrable models”, Nuclear Phys. B, 911 (2016), 902–927 | DOI | MR | Zbl

[19] A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, N. A. Slavnov, “Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 2. Determinant representation”, J. Phys. A, 50:3 (2017), 034004 ; (2016), 22 pp., arXiv: 1606.03573 | DOI

[20] A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, N. A. Slavnov, “Multiple actions of the monodromy matrix in $\mathfrak{gl}(2|1)$-invariant integrable models”, SIGMA, 12 (2016), 099, 22 pp. | DOI | MR

[21] A. Hutsalyuk, A. Liashyk, S. Z. Pakuliak, E. Ragoucy, N. A. Slavnov, “Scalar products of Bethe vectors in models with $\mathfrak{gl}(2|1)$ symmetry 1. Super-analog of Reshetikhin formula”, J. Phys. A, 49:45 (2016), 454005, 27 pp. | DOI | Zbl

[22] V. Chari, A. Pressley, A guide to quantum groups, Cambridge Univ. Press, Cambridge, 1994, xvi+651 pp. | MR | Zbl

[23] V. G. Drinfeld, “Novaya realizatsiya yangianov i kvantovykh affinnykh algebr”, Dokl. AN SSSR, 296:1 (1987), 13–17 ; V. G. Drinfel'd, “A new realization of Yangians and quantized affine algebras”, Soviet Math. Dokl., 36:2 (1988), 212–216 | MR | Zbl

[24] J. Ding, I. B. Frenkel, “Isomorphism of two realizations of quantum affine algebra $U_q(\widehat{\mathfrak{gl}}(N))$”, Comm. Math. Phys., 156:2 (1993), 277–300 | DOI | MR | Zbl

[25] Y.-Z. Zhang, “Super-Yangian double and its central extension”, Phys. Lett. A, 234:1 (1997), 20–26 | DOI | MR | Zbl

[26] B. Enriquez, S. Khoroshkin, S. Pakuliak, “Weight functions and Drinfeld currents”, Comm. Math. Phys., 276:3 (2007), 691–725 ; (2006), 36 pp., arXiv: math/0610398 | DOI | MR | Zbl

[27] L. Frappat, S. Khoroshkin, S. Pakuliak, E. Ragoucy, “Bethe ansatz for the universal weight function”, Ann. Henri Poincaré, 10:3 (2009), 513–548 ; (2009 (v1 – 2008)), 31 pp., arXiv: 0810.3135 | DOI | MR | Zbl

[28] S. Khoroshkin, S. Pakuliak, “Generating series for nested Bethe vectors”, SIGMA, 4 (2008), 081, 23 pp. | DOI | MR | Zbl

[29] A. G. Izergin, “Statisticheskaya summa shestivershinnoi modeli v konechnom ob'eme”, Dokl. AN SSSR, 297:2 (1987), 331–333 | MR | Zbl

[30] S. Pakuliak, E. Ragoucy, N. A. Slavnov, Bethe vectors for models based on the super-Yangian $Y(\mathfrak{gl}(m|n))$, 2016, 30 pp., arXiv: 1604.02311