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Regular version of the site
Of all publications in the section: 23
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Article
Takebe T., Teo L. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2006. Vol. 2. No. 072. P. 1-30.

We define the coupled modified KP hierarchy and its dispersionless limit. This integrable hierarchy is a generalization of the ''half'' of the Toda lattice hierarchy as well as an extension of the mKP hierarchy. The solutions are parametrized by a fibered flag manifold. The dispersionless counterpart interpolates several versions of dispersionless mKP hierarchy.

Added: Aug 13, 2014
Article
Takebe T. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2014. Vol. 10. P. 1-13.

We show that N-variable reduction of the dispersionless BKP hierarchy is described by a Loewner type equation for the quadrant.

Added: Mar 15, 2014
Article
Burman Y. M., Berenstein A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2009. Vol. 5. No. 57. P. 1-18.

Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.

Added: Oct 12, 2012
Article
Khoroshkin S. M., Пакуляк С. З. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2008. Vol. 4. No. 081. P. 23.
Added: Oct 15, 2012
Article
Levin A., Olshanetsky M., Smirnov A. V. et al. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2012. Vol. 8. No. 095.
We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint G-bundles of different topological types over complex curves Sigma(g,n) of genus g with n marked points. The bundles are defined by their characteristic classes - elements of H-2 (Sigma(g,n), Z(G)), where Z (G) is a center of the simple complex Lie group G. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness.
Added: Feb 7, 2013
Article
Nirov Khazret S., Razumov A. V. 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 the expressions for the L-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations.

Added: Jan 29, 2018
Article
Loktev S., Natanzon S. M. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2011. Vol. 7. No. 70. P. 1-15.

We show that any complex (respectively real) representation of finite group naturally generates a open-closed (respectively Klein) topological field theory over complex numbers. We relate the 1-point correlator for the projective plane in this theory with the Frobenius-Schur indicator on the representation. We relate any complex simple Klein TFT to a real division ring.

Added: Oct 12, 2012
Article
Rybnikov L. G., Chervov A., Falqui G. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2009. Vol. 5. No. 29. P. 17.
We consider the XXX homogeneous Gaudin system with N sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals(in the classical case) and new “Gaudin” algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed(in the classical case). We will make use of properties of “Manin matrices” to provide explicit generators of the Gaudin Algebras in the quantum case.
Added: Oct 12, 2012
Article
Spiridonov V., Magadov K. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2018. Vol. 14. No. 121. P. 1-13.

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.  

Added: Nov 14, 2018
Article
Akhtar M., Coates T., Galkin S. et al. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2012. Vol. 8. No. 094. P. 1-707.

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P* (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.

Added: Sep 14, 2013
Article
Zotov A., Levin A., Olshanetsky M. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2009. Vol. 5. No. 065. P. 1-22.
Modifications of bundles over complex curves is an operation that allows one to construct a new bundle from a given one. Modifications can change a topological type of bundle. We describe the topological type in terms of the characteristic classes of the bundle.Being applied to the Higgs bundles modifications establish an equivalence between different classical integrable systems. Following Kapustin and Witten we define the modifications in terms of monopole solutions of the Bogomolny equation. We find the Dirac monopole solution in the case R × (elliptic curve). This solution is a three-dimensional generalization of the Kronecker series. We give two representations for this solution and derive a functional equation for it generalizing the Kronecker results. We use it to define Abelian modifications for bundles of arbitrary rank. We also describe non-Abelian modifications in terms of thetafunctions with characteristic.
Added: Oct 15, 2012
Article
Hutsalyuk A., Liashyk A., Pakuliak S. et al. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2016. Vol. 12. No. 099. P. 1-22.

We study gl(2|1) symmetric integrable models solvable by the nested algebraic Bethe ansatz. Using explicit formulas for the Bethe vectors we derive the actions of the monodromy matrix entries onto these vectors. We show that the result of these actions is a finite linear combination of Bethe vectors. The obtained formulas open a way for studying scalar products of Bethe vectors.  

Added: Nov 17, 2016
Article
Kirillov A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2016. Vol. 12.

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.

Added: May 5, 2016
Article
Ayano T., Nakayashiki A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2013. Vol. 9. No. 046.
Added: Nov 2, 2016
Article
Ayano T. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2016. Vol. 12. No. 086.
Added: Nov 2, 2016
Article
Gavrylenko P., Iorgov N., Lisovyy O. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2018. Vol. 14. P. 1-27.

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for $c=N-1$.

Added: Nov 22, 2018
Article
Ogievetsky O., Khoroshkin S. M. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2011. Vol. 7. No. 64. P. 1-34.
We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra gl n into gl n ⊕gl n . Its representation theory is related to the theory of decompositions of tensor products of gl n -modules.
Added: Oct 15, 2012
Article
Pogrebkov A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2017. Vol. 13. No. 53. P. 1-14.

Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented. Commutativity of these symmetries enables interpretation of their parameters as “times” of the nonlinear integrable partial differential-difference and differential equations. Examples of equations resulting in such procedure and their Lax pairs are given. Besides these, ordinary, symmetries the additional ones are introduced and their action on the Scattering data is presented.

Added: Sep 4, 2017
Article
Sergeev A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2009. Vol. 5. No. 015. P. 1-20.
Added: Feb 19, 2013
Article
Bizyaev I. A., Borisov A. V., Mamaev I. S. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2016. Vol. 12. P. 1-19.

In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them.

Added: Apr 5, 2017
Article
Zabrodin A. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2014. Vol. 10. No. 006. P. 18.

Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages], we show how to construct the master T-operator for the quantum inhomogeneous GL(N) XXX spin chain with twisted boundary conditions. It satisfies the bilinear identity and Hirota equations for the classical mKP hierarchy. We also characterize the class of solutions to the mKP hierarchy that correspond to eigenvalues of the master T-operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum spin chain and the classical Ruijsenaars-Schneider system of particles.

Added: Jul 15, 2014
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