Quadratic algebras related to elliptic curves

We consider a special class of quantum non-dynamical R-matrices in the fundamental representation of GL N with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case N = 2 these are the well-known 6-vertex R-matrix and its 7-vertex deformation. The R-matrices are used for construction of the classical relativistic integrable tops of the Euler-Arnold type. Namely, we describe the Lax pairs with spectral parameter, the inertia tensors and the Poisson structures. The latter are given by the linear Poisson-Lie brackets for the non-relativistic models, and by the classical Sklyanin type algebras in the relativistic cases. In some particular cases the tops are gauge equivalent to the Calogero-Moser-Sutherland or trigonometric Ruijsenaars-Schneider models.

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“…As was mention in [18] (see also [32]) there is a relation between the Lax matrices (3.8) and (3.57). Similarly, to the rational case we have…”

Section: The Case Of Non-standard R-matrixmentioning

confidence: 65%

Trigonometric Integrable Tops from Solutions of Associative Yang–Baxter Equation

Krasnov

Zotov

2019

Ann. Henri Poincaré

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“…This paper is a kind of a brief review of several previous articles [9,12,14,25,27]. The aim is have a fresh approach and some new results by focus attention on possible Lax pairs and a symmetry between relativistic and spectral parameters arizing from the lattice Fourier transformation.…”

Section: The Lax Equationsl (Z) = [L(z) M(z)]mentioning

confidence: 99%

Relativistic elliptic matrix tops and finite Fourier transformations

Zotov

2017

Mod. Phys. Lett. A

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We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the "offshell" Lax pairs, which do not satisfy the Lax equations in general case but become true Lax pairs under various conditions (reductions). At the level of the off-shell Lax matrix there is a natural symmetry between the spectral parameter z and relativistic parameter η. It is generated by the finite Fourier transformation, which we describe in detail. The symmetry allows to consider z and η on an equal footing. Depending on the type of integrable reduction any of the parameters can be chosen to be the spectral one. Then another one is the relativistic deformation parameter. As a by-product we describe the model of N 2 interacting GL(M ) matrix tops and/or M 2 interacting GL(N ) matrix tops depending on a choice of the spectral parameter. Integrable elliptic topsWe consider a special class of the Liouville integrable systems. The simplest example is the (complexified) Euler top inwhere S = (S 1 , S 2 , S 3 ), J( S) = (J 1 S 1 , J 2 S 2 , J 3 S 3 ) and {J i } (inverse components of inertia tensor in principle axes) are arbitrary constants. Equation (1.1) is simply rewritten in matrix form:whereand {σ k } are the Pauli matrices. Equation (1.2) admits generalization to Mat(N, C)-valued S. These model are known as the Euler-Arnold tops [2,3,19,15,20]. They are not integrable for a generic linear constant operator J but only for very specific choices of J.

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“…The construction of the KZB connection is based on the classical dynamical elliptic r-matrix defined as sections of bundles over elliptic curves [13,47,74]. For trivial G-bundles our list coincides with the elliptic r-matrices were defined in [19].…”

Section: Classical R-matrixmentioning

confidence: 99%

“…For arbitrary characteristic classes these type of models were described in [45]. Different aspects and applications of the Hecke transformations to integrable systems and related topics (such as Painlevé-Schlesinger equations [2,15,50,51,60,67,68,71], monopoles [14,25,31,37,39,49,58], quadratic Poisson structures [13,74], applications to AGT conjecture [53,54,55] etc.) can be found in wide range of literature.…”

Section: Introductionmentioning

confidence: 99%

Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

Levin

Olshanetsky

Smirnov

et al. 2012

SIGMA

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Abstract. 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 Σ g,n of genus g with n marked points. The bundles are defined by their characteristic classes -elements of H 2 (Σ 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.

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Quadratic algebras related to elliptic curves

Cited by 20 publications

References 9 publications

Trigonometric Integrable Tops from Solutions of Associative Yang–Baxter Equation

Trigonometric Integrable Tops from Solutions of Associative Yang–Baxter Equation

Relativistic elliptic matrix tops and finite Fourier transformations

Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

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