S. E. Derkachev, A. N. Manashov, “General solution of the Yung–Baxter equation with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$”, Algebra i Analiz, 21:4 (2009), 1–94; St. Petersburg Math. J., 21:4 (2010), 513

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Algebra i Analiz, 2009, Volume 21, Issue 4, Pages 1–94 (Mi aa1145)

This article is cited in 29 scientific papers (total in 29 papers)

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General solution of the Yung–Baxter equation with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$

S. E. Derkachev^a, A. N. Manashov^bc

^a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia
^b Institute for Theoretical Physics, University of Regensburg, Regensburg, Germany
^c St. Petersburg State University, Faculty of Physics, St. Petersburg, Russia

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Abstract: The problem of constructing the $\mathrm R$-matrix is considered in the case of an integrable spin chain with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$. A fairly complete study of general $\mathrm R$-matrices acting in the tensor product of two continuous series representations of $\mathrm{SL}(\mathrm n,\mathbb C)$ is presented. On this basis, $\mathrm R$-matrices are constructed that act in the tensor product of Verma modules (which are infinite-dimensional representations of the Lie algebra $\mathrm{sl}(n)$), and also $\mathrm R$-matrices acting in the tensor product of finite-dimensional representations of the Lie algebra $\mathrm{sl}(n)$.

Received: 19.11.2008

English version:
St. Petersburg Mathematical Journal, 2010, Volume 21, Issue 4, Pages 513–577
DOI: https://doi.org/10.1090/S1061-0022-2010-01106-3

Bibliographic databases:

Document Type: Article

MSC: 81R12

Language: Russian

Citation: S. E. Derkachev, A. N. Manashov, “General solution of the Yung–Baxter equation with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$”, Algebra i Analiz, 21:4 (2009), 1–94; St. Petersburg Math. J., 21:4 (2010), 513–577

Citation in format AMSBIB

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https://www.mathnet.ru/eng/aa1145

https://www.mathnet.ru/eng/aa/v21/i4/p1

This publication is cited in the following 29 articles:

Yury A. Neretin, “On Derkachov–Manashov R-matrices for the principal series of unitary representations”, Journal of Mathematical Physics, 65:5 (2024)
P. V. Antonenko, N. M. Belousov, S. È. Derkachov, P. A. Valinevich, “Reflection operator and hypergeometry II: $SL(2,\mathbb{C})$ spin chain”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 30, Zap. nauchn. sem. POMI, 532, POMI, SPb., 2024, 47–79
N. Belousov, S. Derkachov, “Regular Representation of the Group GL(N, ℝ): Factorization, Casimir Operators and Toda Chain”, J Math Sci, 264:3 (2022), 215
Pavel V Antonenko, “The Gelfand–Tsetlin basis for infinite-dimensional representations of
gln(C)”, J. Phys. A: Math. Theor., 55:22 (2022), 225201
Yury A. Neretin, “Barnes–Ismagilov Integrals and Hypergeometric Functions of the Complex Field”, SIGMA, 16 (2020), 072, 20 pp.
N. M. Belousov, S. E. Derkachev, “Regulyarnoe predstavlenie gruppy $\mathrm{GL}(N,\mathbb{R})$: faktorizatsiya, operatory Kazimira i tsepochka Tody”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 27, Zap. nauchn. sem. POMI, 494, POMI, SPb., 2020, 23–47
S. È. Derkachev, V. P. Spiridonov, “The $6j$-symbols for the $SL(2,\mathbb C)$ group”, Theoret. and Math. Phys., 198:1 (2019), 29–47
P. A. Valinevich, “Construction of the Gelfand–Tsetlin basis for unitary principal series representations of the algebra $sl_n(\mathbb C)$”, Theoret. and Math. Phys., 198:1 (2019), 145–155
Kazakov V., Olivucci E., Preti M., “Generalized Fishnets and Exact Four-Point Correlators in Chiral Cft4”, J. High Energy Phys., 2019, no. 6, 078
Spiridonov V.P., “The Rarefied Elliptic Bailey Lemma and the Yang Baxter Equation”, J. Phys. A-Math. Theor., 52:35 (2019), 355201
Tsuboi Z., “A Note on Q-Oscillator Realizations of U-Q(Gl(M|N)) For Baxter Q-Operators”, Nucl. Phys. B, 947 (2019), UNSP 114747
Chicherin D., Kazakov V., Loebbert F., Mueller D., Zhong D.-l., “Yangian Symmetry For Bi-Scalar Loop Amplitudes”, J. High Energy Phys., 2018, no. 5, 003
Chicherin D., Spiridonov V.P., “The Hyperbolic Modular Double and the Yang-Baxter Equation”, Representation Theory, Special Functions and Painleve Equations - Rims 2015, Advanced Studies in Pure Mathematics, 76, eds. Konno H., Sakai H., Shiraishi J., Suzuki T., Yamada Y., Math Soc Japan, 2018, 95–123
J. Math. Sci. (N. Y.), 242:5 (2019), 658–682
Gahramanov I., Jafarzade Sh., “Integrable Lattice Spin Models From Supersymmetric Dualities”, Phys. Part. Nuclei Lett., 15:6 (2018), 650–667
P. A. Valinevich, S. E. Derkachov, A. P. Isaev, “SOS-representation for the $SL(2,\mathbb C)$-invariant $R$-operator and Feynman diagrams”, J. Math. Sci. (N. Y.), 238:6 (2019), 819–833
Dmitry Chicherin, Sergey E. Derkachov, Vyacheslav P. Spiridonov, “From Principal Series to Finite-Dimensional Solutions of the Yang–Baxter Equation”, SIGMA, 12 (2016), 028, 34 pp.
P. A. Valinevich, S. È. Derkachev, P. P. Kulish, E. M. Uvarov, “Construction of eigenfunctions for a system of quantum minors of the monodromy matrix for an $SL(n,\mathbb C)$-invariant spin chain”, Theoret. and Math. Phys., 189:2 (2016), 1529–1553
Bosnjak G., Mangazeev V.V., “Construction of R -matrices for symmetric tensor representations related to ${U}_{q}(\hat{{{sl}}_{n}})$”, J. Phys. A-Math. Theor., 49:49 (2016), 495204
S. È. Derkachev, V. P. Spiridonov, “Finite-dimensional representations of the elliptic modular double”, Theoret. and Math. Phys., 183:2 (2015), 597–618

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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