Integrability and braided tensor categories by Fendley, P

“…Such currents have been constructed by utilising quantum-group algebras and ideas from “discrete holomorphicity”. …”

On the geometric realization of the inner product and canonical basis for quantum affine $\mathfrak{sl}_n$ by McGerty, K

“…We give a geometric interpretation of the inner product on the modified quantum group of $\hat{\mathfrak{sl}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.…”

Cells in quantum affine sl_n by McGerty, K

“…Using the geometric construction of the quantum group due to Lusztig and Ginzburg--Vasserot, we describe explicitly the two-sided cells, the number of left cells in a two--sided cell, and the asymptotic algebra, verifying conjectures of Lusztig.…”

Universal Bethe ansatz solution for the Temperley–Lieb spin chain by Rafael I. Nepomechie, Rodrigo A. Pimenta

“…We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. …”
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The Robinson―Schensted Correspondence and $A_2$-webs by Matthew Housley, Heather M. Russell, Julianna Tymoczko

“…The $A_2$-spider category encodes the representation theory of the $sl_3$ quantum group. Kuperberg (1996) introduced a combinatorial version of this category, wherein morphisms are represented by planar graphs called $\textit{webs}$ and the subset of $\textit{reduced webs}$ forms bases for morphism spaces. …”
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An Ising-type formulation of the six-vertex model by Vladimir V. Bazhanov, Sergey M. Sergeev

“…The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. …”
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Enhancement of InGaN Quantum Well Photoluminescence in a Tamm Metal/Porous-DBR Micro-Cavity by Andrei Sarua, Jon Pugh, Edmund Harbord, Martin Cryan

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Resource theories of communication by Hlér Kristjánsson, Giulio Chiribella, Sina Salek, Daniel Ebler, Matthew Wilson

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Signature Characters of Highest-Weight Representations of U[subscript q](𝖌l[subscript n] by Venkateswaran, Vidya

“…We consider U[subscript q](𝖌l[subscript n), the quantum group of type A for |q| = 1, q generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. …”
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Quantum twist-deformed D = 4 phase spaces with spin sector and Hopf algebroid structures by Jerzy Lukierski, Stjepan Meljanac, Mariusz Woronowicz

“…Two Hopf algebroid structures of generalized phase spaces with spin sector will be investigated: first one H(10,10) describing dynamics on quantum group algebra Gˆ provides by the Heisenberg double algebra HD=H⋊Gˆ, and second, denoted by H˜(10,10), describing twisted Hopf algebroid with base space containing twisted noncommutative Minkowski space xˆμ. …”
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Onsager symmetries in $U(1)$ -invariant clock models by Vernier, E, O'Brien, E, Fendley, P

“…We construct the elements of the algebra explicitly from transfer matrices built from non-fundamental representations of the quantum-group algebra . We analyse the spectra further by using both the coordinate Bethe ansatz and a functional approach, and show that the degeneracies result from special exact n-string solutions of the Bethe equations. …”

Affine quantum algebras, Weyl groups and constructible functions by McGerty, Kevin (Kevin Rory), 1975-

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Symmetric elliptic functions, IRF models, and dynamic exclusion processes by Borodin, Alexei

“…© European Mathematical Society 2020 We introduce stochastic Interaction-Round-a-Face (IRF) models that are related to representations of the elliptic quantum group Eτ,η(sl2). For stochastic IRF models in a quadrant, we evaluate averages for a broad family of observables that can be viewed as higher analogs of q-moments of the height function for the stochastic (higher spin) six vertex models. …”
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Modified Turaev-Viro invariants from quantum 𝔰𝔩(2|1) by Anghel, CA, Geer, N

“…Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize q to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. …”

Duality Theory and Categorical Universal Logic: With Emphasis on Quantum Structures by Yoshihiro Maruyama

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String-net formulation of Hamiltonian lattice Yang-Mills theories and quantum many-body scars in a nonabelian gauge theory by Tomoya Hayata, Yoshimasa Hidaka

“…Following the string-net model, we introduce a regularization of the Kogut-Susskind Hamiltonian of lattice Yang-Mills theory based on the q deformation, which respects the (discretized) SU(2) gauge symmetry as quantum group, i.e., SU(2) k , and enables implementation of the lattice Yang-Mills theory both in classical and quantum algorithms by referring to those of the string-net model. …”
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Quantum spectral curve for the η-deformed AdS5ÃS5 superstring by Rob Klabbers, Stijn J. van Tongeren

“…We discuss how the same concepts apply to the η-deformed AdS5ÃS5 superstring, an integrable deformation of the AdS5ÃS5 superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the AdS5ÃS5 superstring, like the relation between the XXZ and XXX spin chains, or the sausage and the S2 sigma models for instance. …”
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The Real Forms of the Fractional Supergroup SL(2,C) by Yasemen Ucan, Resat Kosker

“…There are real forms of the classical Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> and the quantum group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> in the literature. …”
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Weak Multiplier Hopf Algebras II: Source and Target Algebras by Alfons Van Daele, Shuanhong Wang

“…We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known ’quantization’ of the groupoid that comes from an action of a group on a set. …”
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