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## Quantum groups and functional relations for lower rank

Journal of Geometry and Physics. 2017. Vol. 112. P. 1-28.

Nirov Khazret S., Razumov A. V.

A detailed construction of the universal integrability objects related to the integrable

systems associated with the quantum loop algebra Uq(L(sl2)) is given. The full proof of the

functional relations in the form independent of the representation of the quantum loop

algebra on the quantum space is presented. The case of the general gradation and general

twisting is treated. The specialization of the universal functional relations to the case when

the quantum space is the state space of a discrete spin chain is described. This is a digression

of the corresponding consideration for the case of the quantum loop algebra Uq(L(sl3)) with

an extension to the higher spin case.

Boos H., Göhmann F., Klümper A. et al., Journal of Mathematical Physics 2016 Vol. 57 No. 111702 P. 0-23

We find the ℓ-weights and the corresponding ℓ-weight vectors for the finite and
infinite dimensional representations of the quantum loop algebras Uq(L(sl2)) and
Uq(L(sl3)) obtained from the Verma representations of the quantum groups Uq(gl2)
and Uq(gl3) via the Jimbo’s homomorphism. Then we find the ℓ-weights and the
ℓ-weight vectors for the q-oscillator representations of the positive Borel subalgebras
of the ...

Added: January 29, 2018

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2019 No. 074003 P. 1-22

We establish the exact laws of large numbers for two time additive quantities in the raise and peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the ...

Added: October 8, 2019

Nirov Khazret S., Razumov A. V., Journal of Physics: Conference Series 2016 Vol. 670 No. 012037 P. 0-17

By the universal integrability objects we mean certain monodromy-type and transfer-type
operators, where the representation in the auxiliary space is properly fixed, while the
representation in the quantum space is not. This notion is actually determined by the structure of
the universal R-matrix. We call functional relations between such universal integrability objects,
and so, being independent of the representation ...

Added: January 29, 2018

Galkin S., Belmans P., Mukhopadhyay S., / Cornell University. Series math "arxiv.org". 2020. No. 2009.05568.

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode degenerations of the moduli space of rank 2 bundles with fixed determinant. We show that the birational type of the graph potential only depends on the homotopy type of ...

Added: April 15, 2021

Marshall I., International Mathematics Research Notices 2015 Vol. 18 P. 8925-8958

A Poisson structure is defined on the space {\mathcal {W}} of twisted polygons in {\mathbb {R}}^{\nu }. Poisson reductions with respect to two Poisson group actions on {\mathcal {W}} are described. The \nu =2 and \nu =3 cases are discussed in detail. Amongst the Poisson structures arising in examples are to be found the lattice ...

Added: November 28, 2014

Marshakov A., Fock V., / Cornell University. Series math "arxiv.org". 2014.

We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups PGLˆ(N), which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups (Wˆ×Wˆ)♯. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the ...

Added: October 29, 2014

Kharchev S., Levin A., Olshanetsky M. et al., Journal of Mathematical Physics 2018 Vol. 59 No. 103509 P. 1-36

We define the quasi-compact Higgs G -bundles over singular curves introduced in our previous paper for the Lie group SL(N). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of G at marked points of the curves. We demonstrate that in particular cases, this construction leads ...

Added: October 20, 2018

Boos H., Göhmann F., Klümper A. et al., Journal of Mathematical Physics 2017 Vol. 58 No. 093504 P. 093504-1-093504-23

We find the ℓ-weights and the ℓ-weight vectors for the highest ℓ-weight q-oscillator representations
of the positive Borel subalgebra of the quantum loop algebra Uq(L(sll+1))
for arbitrary values of l. Having this, we establish the explicit relationship between the
q-oscillator and prefundamental representations. Our consideration allows us to conclude
that the prefundamental representations can be obtained by tensoring q-oscillator
representations. ...

Added: January 29, 2018

Kamnitzer J., Halacheva I., Weekes A. et al., / Cornell University. Series math "arxiv.org". 2017.

Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable genus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the ...

Added: October 17, 2017

Nirov Khazret S., Razumov A., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2017 Vol. 13 No. 043 P. 1-31

We discuss highest ℓ-weight representations of quantum loop algebras and the
corresponding functional relations between integrability objects. In particular, we compare
the prefundamental and q-oscillator representations of the positive Borel subalgebras of
the quantum group Uq(L(sll+1)) for arbitrary values of l. Our article has partially the
nature of a short review, but it also contains new results. These are ...

Added: January 29, 2018

Derbyshev A. E., Povolotsky A. M., Priezzhev V. B., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2015 Vol. 91 P. 022125

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. (2012) P05014] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as ...

Added: February 19, 2015

Kazaryan M., Lando S., Успехи математических наук 2015 Т. 70 № 3 С. 70-106

This paper reviews modern approaches to the construction of formal solutions to integrable hierarchies of mathematical physics whose coefficients are answers to various enumerative problems. The relationship between these approaches and the combinatorics of symmetric groups and their representations is explained. Applications of the results to the construction of efficient computations in problems related to ...

Added: September 21, 2015

Khoroshkin S. M., Tsuboi Z., Journal of Physics A: Mathematical and Theoretical 2014 Vol. 47 P. 1-11

We consider the 'universal monodromy operators' for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the Uq(\hat{sl}(2)) case. ...

Added: December 8, 2014

Klümper A., Nirov K., Razumov A., Journal of Physics A: Mathematical and Theoretical 2020 Vol. 53 No. 1 P. 1-35

We use the quantum group approach for the investigation of correlation functions
of integrable vertex models and spin chains. For the inhomogeneous
reduced density matrix in case of an arbitrary simple Lie algebra we find
functional equations of the form of the reduced quantum Knizhnik–
Zamolodchikov equation. This equation is the starting point for the
investigation of correlation functions at ...

Added: January 26, 2021

Buryak A., Dubrovin B., Guere J. et al., International Mathematics Research Notices 2020 Vol. 2020 No. 24 P. 10381-10446

In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus 1 quantum correction and, as an application, compute completely the quantization ...

Added: April 21, 2020

Krichever I. M., Функциональный анализ и его приложения 2012 Т. 46 № 2 С. 37-51

Using meromorphic differentials with real periods, we prove Arbarello's conjecture that any compact complex cycle of dimension g−n in the moduli space M_g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n. ...

Added: April 17, 2014

Buryak A., Rossi P., Advances in Mathematics 2021 Vol. 386 No. 6 Article 107794

In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's r-spin classes. They are parameterized by a phase space which has one extra dimension and in genus 0 they correspond to the extended r-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while ...

Added: October 29, 2021

Marshakov A., Семенякин Н. С., Journal of High Energy Physics 2019 Vol. 100 No. 10 P. 1-52

We discuss relation between the cluster integrable systems and spin chains in the context of their correspondence with 5d supersymmetric gauge theories. It is shown that glN XXZ-type spin chain on M sites is isomorphic to a cluster integrable system with N × M rectangular Newton polygon and N × M fundamental domain of a ...

Added: October 21, 2019

Buryak A., Rossi P., Communications in Mathematical Physics 2016 Vol. 342 No. 2 P. 533-568

In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced by the first author using intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the ...

Added: September 28, 2020

A. Zabrodin, A. Zotov, Nuclear Physics B 2018 Vol. 927 P. 550-565

We discuss a self-dual form or the Backlund transformations for the continuous (in time variable) glN Ruijsenaars-Schneider model. It is based on the first order equations in N+M complex variables which include N positions of particles and M dual variables. The latter satisfy equations of motion of the glM Ruijsenaars-Schneider model. In the elliptic case ...

Added: February 15, 2018

Takebe T., Tokyo : Research Center for Mathematical Physics, Rikkyo Universty, 2014

This is a lecture note based on the series of lectures on the dispersionless integrable hierarchies delivered by the authore in June, 2013, at the Rikkyo University, Tokyo, Japan. The contents are survey on dispersionless integrable hierarchies, including introduction to integrable systems in general, and on their connections with complex analysis. ...

Added: June 21, 2014

Marshakov A., Journal of Geometry and Physics 2012 Vol. 003 P. 16-36

We discuss the Poisson structures on Lie groups and propose an explicit construction of the integrable models on their appropriate Poisson submanifolds. The integrals of motion for the SL(N)-series are computed in cluster variables via the Lax map. This construction, when generalised to the co-extended loop groups, gives rise not only to alternative descriptions of relativistic Toda systems, but allows ...

Added: February 11, 2013

Marshakov A., International Journal of Modern Physics A 2013 Vol. 28 No. 3-4 P. 1340007

We propose an explicit construction for the integrable models on Poisson submanifolds of the Lie groups. The integrals of motion are computed in cluster variables via the Lax map. This generalized construction for the co-extended loop groups allows to formulate, in general terms, some new classes of integrable models. ...

Added: March 28, 2013

Povolotsky A. M., Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 46 P. 465205

The conditions of the integrability of general zero range chipping models with factorized steady states, which were proposed in Evans et al (2004 J. Phys. A: Math. Gen. 37 L275), are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle ...

Added: November 14, 2013