Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 068, 67 pages      arXiv:1811.09401      https://doi.org/10.3842/SIGMA.2019.068

Vertex Models and Spin Chains in Formulas and Pictures

Khazret S. Nirov abc and Alexander V. Razumov d
a) Institute for Nuclear Research of the Russian Academy of Sciences, 7a 60th October Ave., 117312 Moscow, Russia
b) Faculty of Mathematics, National Research University ''Higher School of Economics'', 119048 Moscow, Russia
c) Mathematics and Natural Sciences, University of Wuppertal, 42097 Wuppertal, Germany
d) NRC ''Kurchatov Institute — IHEP'', 142281 Protvino, Moscow region, Russia

Received March 19, 2019, in final form August 30, 2019; Published online September 13, 2019

Abstract
We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are introduced. Their explicit analytical forms for the case of integrable systems associated with the quantum loop algebra ${\mathrm U}_q(\mathcal L(\mathfrak{sl}_{l + 1}))$ are given. The commutativity conditions for the transfer operators of lattices with a boundary are derived by the graphical method. Our consideration reveals useful advantages of the graphical approach for certain problems in the theory of quantum integrable systems.

Key words: quantum loop algebras; integrable vertex models; integrable spin models; graphical methods; open chains.

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