Semi-stable vector bundles on elliptic curves and the associative Yang–Baxter equation

Burban, Igor; Henrich, Thilo

doi:10.1016/j.geomphys.2011.11.001

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…The aim of this section is to use the main results from [6] and the construction above, in order to obtain new solutions of the Rota-Baxter equation with spectral parameters. As the title suggest, these comes from the world of vector bundles on elliptic curves.…”

Section: Elliptic Curves and The Associative Yang-baxter Equation Witmentioning

confidence: 99%

On a class of Rota-Baxter operators with geometric origin

Anghel¹

2019

AIP Conference Proceedings

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Section: Elliptic Curves and The Associative Yang-baxter Equation Witmentioning

confidence: 99%

On a class of Rota-Baxter operators with geometric origin

Anghel¹

2019

AIP Conference Proceedings

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“…The most natural 6 and simple dynamical solution of (1.15) was proposed by Burban and Henrich [9] (see also [21]):…”

Section: Brief Reviewmentioning

confidence: 99%

Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

Сечин

Zotov

2016

Journal of Mathematical Physics

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In this paper we propose versions of the associative Yang-Baxter equation and higher order R-matrix identities which can be applied to quantum dynamical R-matrices. As is known quantum non-dynamical R-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum YangBaxter equation and a set of identities useful for different applications in integrable systems. The dynamical R-matrices satisfy the Gervais-Neveu-Felder (or dynamical Yang-Baxter) equation. Relation between the dynamical and non-dynamical cases is described by the IRF-Vertex transformation. An alternative approach to quantum (semi-)dynamical Rmatrices and related quantum algebras was suggested by Arutyunov, Chekhov and Frolov (ACF) in their study of the quantum Ruijsenaars-Schneider model. The purpose of this paper is twofold. First, we prove that the ACF elliptic R-matrix satisfies the associative Yang-Baxter equation with shifted spectral parameters. Second, we directly prove a simple relation of the IRF-Vertex type between the Baxter-Belavin and the ACF elliptic R-matrices predicted previously by Avan and Rollet. It provides the higher order R-matrix identities and an explanation of the obtained equations through those for non-dynamical R-matrices. As a by-product we also get an interpretation of the intertwining transformation as matrix extension of scalar theta function likewise R-matrix is interpreted as matrix extension of the Kronecker function. Relations to the Gervais-Neveu-Felder equation and identities for the Felder's elliptic R-matrix are also discussed.

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“…Belavin and Drinfeld worked on solutions of the classical Yang-Baxter equation for simple Lie algebras [5]. Burban and Henrich handled semi-stable vector bundles on elliptic curves and their relation with associative Yang-Baxter equation [26]. Nichita and Parashar studied Spectral-parameter dependent Yang-Baxter operators and Yang-Baxter systems from algebraic structures [12], and etc.…”

Section: Introductionmentioning

confidence: 99%

Geometrical approach on set theoretical solutions of Yang-Baxter equation in Lie algebras

Bozdağ

Senturk

2022

Malaya J. Mat.

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In this paper, we handle set-theoretical solutions of Yang-Baxter equation and Lyubashenko set theoretical solutions in Lie algebras. We present a new commutative binary operation on these structures, and we obtain new set theoretical solutions including this operation by using property of commutativity of it. Also, we show that some set theoretical solutions of Yang-Baxter equation corresponds to the Lyubashenko set theoretical solutions on these structures. Additionally, we give some relations to verify set theoretical solution of Yang-Baxter equation. Moreover, we put an interpretation for these solutions from the point of geometrical view in Euclidean space, Minkowski space and differentiable manifolds by using Lie algebras.

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Semi-stable vector bundles on elliptic curves and the associative Yang–Baxter equation

Cited by 4 publications

References 16 publications

On a class of Rota-Baxter operators with geometric origin

On a class of Rota-Baxter operators with geometric origin

Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

Geometrical approach on set theoretical solutions of Yang-Baxter equation in Lie algebras

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