B. De-gier, K. Nienhuis, M. W. Nirov, T. Bautista, D. Betea et al., It was a pleasure for me to work with them and it will remain to be so in the future. I benefited from inspiring discussions with Valentin Verschinin (for discussing Super Yang?Mills theories), Michael Wheeler and Jean-Bernard Zuber. I would like to thank Vladimir Bazhanov and Michio Jimbo for the discussions of the Q-operators. A large part of this thesis was written during the workshop " Statistical mechanics, integrability and combinatorics " in the Galileo Galilei Institute for Theoretical Physics (GGI) in Florence. I would like to thank GGI for the hospitality and the INFN for partial support. I would like to thank Filippo Colomo, the local organiser, who took care of many practical issues and was always very helpful. I am also grateful to the organisers of the summer schools in Hamburg, Germany " Quantum Groups and Integrability ; Algebraic, Analytic and Geometric Aspects " and Turunç, Turkey " Exact and Numerical Models of Low-Dimensional Quantum Structures " . During my PhD I visited a number of times the University of Amsterdam and the University of Melbourne to which I express my gratitude for their hospitality and support. Special thanks to Peter Finch for reading my thesis on an earlier stage and pointing to a number of deficiencies. The comments of my rapporteurs Nikolai Kitnanine and Khazret Nirov are greatly appreciated. During my PhD years I was morally supported by Elena Panfilova, my parents and my brother. I am very grateful to them. I would like to thank my fellow PhD students and postdocs Anita, an independent researcher. I would like to thank my collaborators George Féher Thibault and Tianhan, for creating a worm and friendly atmosphere in and " out

M. T. Batchelor, J. De-gier, and B. Nienhuis, chain at ?? = -??, alternating-sign matrices and plane partitions, Journal of Physics A: Mathematical and General, vol.34, issue.19, pp.265-270, 2001.
DOI : 10.1088/0305-4470/34/19/101

R. J. Baxter, Exactly Solved Models in Statistical Mechanics, 1982.
DOI : 10.1142/9789814415255_0002

R. J. Baxter, The Six and Eight-Vertex Models Revisited, Journal of Statistical Physics, vol.116, issue.1-4, pp.43-66, 2004.
DOI : 10.1023/B:JOSS.0000037215.07702.93

V. V. Bazhanov, A. N. Hibberd, and S. M. Khoroshkin, Integrable structure of Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory, Nuclear Physics B, vol.622, issue.3, pp.475-574, 2002.
DOI : 10.1016/S0550-3213(01)00595-8

V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Integrable Structure of Conformal Field Theory II. Q-operator and DDV equation, Communications in Mathematical Physics, vol.190, issue.2, pp.247-278, 1997.
DOI : 10.1007/s002200050240

V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Spectral Determinants for Schrödinger Equation and Q-Operators of Conformal Field Theory, Journal of Statistical Physics, vol.102, pp.3-4567, 2001.

V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Higher-level eigenvalues of Q-operators and Schrodinger equation, Advances in Theoretical and Mathematical Physics, vol.7, issue.4, pp.711-725, 2003.
DOI : 10.4310/ATMP.2003.v7.n4.a4

V. V. Bazhanov and Z. Tsuboi, Baxter's Q-operators for supersymmetric spin chains, Nuclear Physics B, vol.805, issue.3, pp.451-516, 2008.
DOI : 10.1016/j.nuclphysb.2008.06.025

J. Beck, Braid group action and quantum affine algebras, Communications in Mathematical Physics, vol.117, issue.3, pp.555-568, 1994.
DOI : 10.1007/BF02099423

URL : http://arxiv.org/abs/hep-th/9404165

R. E. Behrend and V. A. Knight, Higher spin alternating sign matrices, Electr. J. Comb, vol.14, issue.1, 2007.

D. Betea and M. Wheeler, Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices, Journal of Combinatorial Theory, Series A, vol.137, 2014.
DOI : 10.1016/j.jcta.2015.08.007

D. Betea, M. Wheeler, and P. Zinn-justin, Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures, Journal of Algebraic Combinatorics, vol.38, issue.3, 2014.
DOI : 10.1007/s10801-015-0592-3

URL : https://hal.archives-ouvertes.fr/hal-01150572

H. Bethe, Zur Theorie der Metalle Zeitschrift für Physik A Hadrons and Nuclei, pp.205-226, 1931.

H. Boos, F. Göhmann, A. Klümper, K. S. Nirov, and A. V. Razumov, -matrix, Journal of Physics A: Mathematical and Theoretical, vol.43, issue.41, p.415208, 2010.
DOI : 10.1088/1751-8113/43/41/415208

URL : https://hal.archives-ouvertes.fr/in2p3-00459276

H. Boos, F. Göhmann, A. Klümper, K. S. Nirov, and A. V. Razumov, On the universal $\mbox{\boldmath $R$}$-matrix for the Izergin???Korepin model, Journal of Physics A: Mathematical and Theoretical, vol.44, issue.35, p.355202, 2011.
DOI : 10.1088/1751-8113/44/35/355202

H. Boos, F. Göhmann, A. Klümper, K. S. Nirov, and A. V. Razumov, Universal integrability objects, Theoretical and Mathematical Physics, vol.39, issue.1, pp.21-39
DOI : 10.1007/s11232-013-0002-8

URL : http://arxiv.org/abs/1205.4399

H. Boos, F. Göhmann, A. Klümper, K. S. Nirov, and A. V. Razumov, Quantum groups and functional relations for higher rank, Journal of Physics A: Mathematical and Theoretical, vol.47, issue.27, p.275201, 2014.
DOI : 10.1088/1751-8113/47/27/275201

L. Cantini, Finite size emptiness formation probability of the XXZ spin chain at

L. Cantini and A. Sportiello, Proof of the Razumov???Stroganov conjecture, Journal of Combinatorial Theory, Series A, vol.118, issue.5, pp.1549-1574, 2011.
DOI : 10.1016/j.jcta.2011.01.007

A. Caradoc, O. Foda, and N. Kitanine, Higher spin vertex models with domain wall boundary conditions, Journal of Statistical Mechanics: Theory and Experiment, vol.2006, issue.03, p.3012, 2006.
DOI : 10.1088/1742-5468/2006/03/P03012

URL : https://hal.archives-ouvertes.fr/hal-00171157

R. Carter, Lie algebras of finite and ane type, 2005.

J. Caux, algorithm, Journal of Mathematical Physics, vol.50, issue.9, 2009.
DOI : 10.1063/1.3216474

URL : https://hal.archives-ouvertes.fr/hal-01098546

J. Caux and P. Calabrese, Dynamical density-density correlations in the one-dimensional Bose gas, Physical Review A, vol.74, issue.3, p.74031605, 2006.
DOI : 10.1103/PhysRevA.74.031605

V. Chari and A. Pressley, Quantum affine algebras, Communications in Mathematical Physics, vol.117, issue.2, pp.261-283, 1991.
DOI : 10.1007/BF02102063

V. Chari and A. Pressley, Quantum Ane Algebras and their Representations. arXiv:hep- th/9411145v1, 1994.

V. Chari and A. Pressley, A guide to quantum groups, 1995.

V. Chari and A. Pressley, Twisted Quantum Affine Algebras, Communications in Mathematical Physics, vol.196, issue.2, pp.461-476, 1998.
DOI : 10.1007/s002200050431

URL : http://arxiv.org/abs/q-alg/9611002

F. Colomo and A. G. Pronko, The Limit Shape of Large Alternating Sign Matrices, SIAM Journal on Discrete Mathematics, vol.24, issue.4, pp.1558-1571, 2010.
DOI : 10.1137/080730639

F. Colomo and A. G. Pronko, The Arctic Curve of the Domain-Wall Six-Vertex Model, Journal of Statistical Physics, vol.50, issue.5, pp.662-700, 2010.
DOI : 10.1007/s10955-009-9902-2

URL : https://hal.archives-ouvertes.fr/hal-00526718

F. Colomo and A. G. Pronko, Thermodynamics of the Six-Vertex Model in an L-Shaped Domain, Communications in Mathematical Physics, vol.104, issue.3, pp.1-30, 2015.
DOI : 10.1007/s00220-015-2406-9

F. Colomo, A. G. Pronko, and P. Zinn-justin, The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime, J. Stat. Mech. Theory Exp, p.3002, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00526718

I. Damiani, The R-matrix for (twisted) ane quantum algebras. Representations and quantizations , China High, 2000.

J. De-gier, B. Nienhuis, and A. Ponsaing, Exact spin quantum Hall current between boundaries of a lattice strip, Nuclear Physics B, vol.838, issue.3, pp.371-390, 2010.
DOI : 10.1016/j.nuclphysb.2010.05.019

J. Ding and S. M. Khoroshkin, Universal R-matrix for quantum ane algebras U q (A (2) 2 ) and

. Uq, osp(1|2)) with Drinfeld comultiplication, Advances in Mathematics, vol.189, issue.2, pp.413-438, 2004.

R. K. Dodd and R. K. Bullough, Polynomial Conserved Densities for the Sine-Gordon Equations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.352, issue.1671, pp.481-503, 1977.
DOI : 10.1098/rspa.1977.0012

P. Dorey, C. Dunning, and R. Tateo, The ODE/IM correspondence, Journal of Physics A: Mathematical and Theoretical, issue.32, pp.40-205, 2007.

P. Dorey, S. Faldella, S. Negro, and R. Tateo, The Bethe ansatz and the Tzitzeica-Bullough-Dodd equation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.4, issue.5, pp.371-2013, 1989.
DOI : 10.1103/PhysRevLett.69.710

P. Dorey and R. Tateo, Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations, Journal of Physics A: Mathematical and General, vol.32, issue.38, p.419, 1999.
DOI : 10.1088/0305-4470/32/38/102

V. G. Drinfeld, A new realization of Yangians and quantized ane algebras, Soviet Math. Dokl, vol.36, pp.212-216, 1988.

N. Fabbri, M. Panfil, D. Clément, L. Fallani, M. Inguscio et al., Dynamical structure factor of one-dimensional Bose gases: Experimental signatures of beyond-Luttinger-liquid physics, Physical Review A, vol.91, issue.4, p.43617, 2015.
DOI : 10.1103/PhysRevA.91.043617

L. D. Faddeev, E. K. Sklyanin, and L. A. Takhtajan, Quantum inverse problem method I, Theor. Math. Phys, vol.40, pp.688-70686, 1979.

G. Fehér and B. Nienhuis, Currents in the dilute O(n = 1) loop model

D. Fioravanti, F. Ravanini, and M. Stanishkov, Generalized KdV and quantum inverse scattering description of conformal minimal models, Physics Letters B, vol.367, issue.1-4, pp.113-120, 1996.
DOI : 10.1016/0370-2693(95)01463-2

T. Fonseca and P. Zinn-justin, On some ground state components of the O(1) loop model Journal of Statistical Mechanics: Theory and Experiment, P03025, 2009. [45] T. Fonseca and P. Zinn-Justin. Higher Spin Polynomial Solutions of Quantum Knizhnik? Zamolodchikov Equation. Communications in Mathematical Physics, pp.1079-1115, 2009.

D. Francesco, Inhomogeneous loop models with open boundaries, Journal of Physics A: Mathematical and General, vol.38, issue.27, p.6091, 2005.
DOI : 10.1088/0305-4470/38/27/001

D. Francesco and P. Zinn-justin, Around the Razumov Stroganov Conjecture: Proof of a Multi-Parameter Sum Rule, The Electronic Journal of Combinatorics, vol.12, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00004601

D. Francesco and P. Zinn-justin, The quantum Knizhnik???Zamolodchikov equation, generalized Razumov???Stroganov sum rules and extended Joseph polynomials, Journal of Physics A: Mathematical and General, vol.38, issue.48, p.815, 2005.
DOI : 10.1088/0305-4470/38/48/L02

D. Francesco and P. Zinn-justin, Quantum Knizhnik???Zamolodchikov equation: reflecting boundary conditions and combinatorics, Journal of Statistical Mechanics: Theory and Experiment, vol.2007, issue.12, p.12009, 2007.
DOI : 10.1088/1742-5468/2007/12/P12009

URL : https://hal.archives-ouvertes.fr/hal-00203605

P. , D. Francesco, P. Zinn-justin, and J. Zuber, Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain, J. Stat. Mech, p.8011, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00020303