Simple vector bundles on a nodal Weierstrass cubic and quasi-trigonometric solutions of the classical Yang–Baxter equation

Burban, Igor; Galinat, Lennart; Stolin, Alexander

doi:10.1088/1751-8121/aa8e35

Cited by 6 publications

(11 citation statements)

References 31 publications

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“…to the identity (15). It follows that x, a i 1 ...i n ] ∈ J for any x ∈ L, implying that J is an ideal in L. However, L does not contain any non-zero finite-dimensional ideals; see [31,Lemma 8.6].…”

Section: Basic Facts On Affine Lie Algebrasmentioning

confidence: 99%

“…We define a nodal Weierstraß curve E via the pushout diagram (76). We recall now the description of the sheaf A given in [15,Proposition 3.3] (see also [17,Section 5.1.2]).…”

Section: Consider the Embedding Of Lie Algebras Gθmentioning

confidence: 99%

“…Let E be nodal. The (quasi-)trigonometric solution r trg (c,d) (x, y) of ( 36) was computed in [15,Theorem A]. We recall the corresponding formula.…”

Section: Consider the Embedding Of Lie Algebras Gθmentioning

confidence: 99%

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Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

Abedin

Burban

2021

Commun. Math. Phys.

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This paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang–Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang–Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples.

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Section: Basic Facts On Affine Lie Algebrasmentioning

confidence: 99%

“…We define a nodal Weierstraß curve E via the pushout diagram (76). We recall now the description of the sheaf A given in [15,Proposition 3.3] (see also [17,Section 5.1.2]).…”

Section: Consider the Embedding Of Lie Algebras Gθmentioning

confidence: 99%

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Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

Abedin

Burban

2021

Commun. Math. Phys.

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“…Distinguished quasi-trigonometric solutions of the classical Yang-Baxter equation are found in [38]. The q-deformation of maximally extended sl(2|2) is investigated by means of a contraction limit in [39].…”

Section: Quantum Groupsmentioning

confidence: 99%

Quantum integrability and quantum groups: a special issue in memory of Petr P Kulish

Kitanine

Nepomechie

Reshetikhin

2018

J. Phys. A: Math. Theor.

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“…• We answer questions one and two posed at the end of [8] concerning an explicit formula for the quasi-trigonometric solution given by a BD quadruple Q and its connection with the trigonometric solution described by the same quadruple Q (see [3]).…”

Section: Introductionmentioning

confidence: 99%

Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

Abedin,

Maximov

2020

Preprint

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The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE.

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Simple vector bundles on a nodal Weierstrass cubic and quasi-trigonometric solutions of the classical Yang–Baxter equation

Abstract: In this paper we study the combinatorics of quasi-trigonometric solutions of the classical Yang-Baxter equation, arising from simple vector bundles on a nodal Weierstraß cubic.

Cited by 6 publications

References 31 publications

Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

Quantum integrability and quantum groups: a special issue in memory of Petr P Kulish

Classification of classical twists of the standard Lie bialgebra structure on a loop algebra

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