Duality for Knizhnik–Zamolodchikov and Dynamical Operators
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the Knizhnik–Zamolodchikov and dynamical operators, both differential and difference, in the context of the $(\mathfrak{gl}_{k}, \mathfrak{gl}_{n})$-duality for the space of polynomials in $kn$ anticommuting variables. We show that the Knizhnik–Zamolodchikov and dynamical operators naturally exchange under the duality.
Keywords: Knizhnik–Zamolodchikov operators, dynamical operators, the $(\mathfrak{gl}_{k}, \mathfrak{gl}_{n})$-duality.
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     title = {Duality for {Knizhnik{\textendash}Zamolodchikov} and {Dynamical} {Operators}},
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Vitaly Tarasov; Filipp Uvarov. Duality for Knizhnik–Zamolodchikov and Dynamical Operators. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a34/

[1] Cheng S.-J., Wang W., Dualities and representations of Lie superalgebras, Graduate Studies in Mathematics, 144, Amer. Math. Soc., Providence, RI, 2012 | DOI | MR | Zbl

[2] Etingof P., Frenkel I., Kirillov A., Lectures on representation theory and Knizhnik–Zamolodchikov equations, Mathematical Surveys and Monographs, 58, Amer. Math. Soc., Providence, RI, 1998 | DOI | MR | Zbl

[3] Etingof P., Varchenko A., “Dynamical Weyl groups and applications”, Adv. Math., 167 (2002), 74–127, arXiv: math.QA/0011001 | DOI | MR | Zbl

[4] Felder G., Markov Y., Tarasov V., Varchenko A., “Differential equations compatible with KZ equations”, Math. Phys. Anal. Geom., 3 (2000), 139–177, arXiv: math.QA/0001184 | DOI | MR | Zbl

[5] Tarasov V., Varchenko A., “Difference equations compatible with trigonometric KZ differential equations”, Int. Math. Res. Not., 2000 (2000), 801–829, arXiv: math.QA/0002132 | DOI | MR | Zbl

[6] Tarasov V., Varchenko A., “Duality for Knizhnik–Zamolodchikov and dynamical equations”, Acta Appl. Math., 73 (2002), 141–154, arXiv: math.QA/0112005 | DOI | MR | Zbl

[7] Tarasov V., Varchenko A., “Dynamical differential equations compatible with rational qKZ equations”, Lett. Math. Phys., 71 (2005), 101–108, arXiv: math.QA/0403416 | DOI | MR | Zbl

[8] Toledano-Laredo V., “The trigonometric Casimir connection of a simple Lie algebra”, J. Algebra, 329 (2011), 286–327, arXiv: 1003.2017 | DOI | MR | Zbl

[9] Toledano-Laredo V., Yang Y., The elliptic Casimir connection of a simple Lie algebra, arXiv: 1805.12261 | MR

[10] Vicedo B., Young C., “$(\mathfrak{gl}_M,\mathfrak{gl}_N)$-dualities in Gaudin models with irregular singularities”, SIGMA, 14 (2018), 040, 28 pp., arXiv: 1710.08672 | DOI | MR | Zbl