Non-Commutative Rational Yang–Baxter Maps

Doliwa, Adam

doi:10.1007/s11005-013-0669-7

Cited by 30 publications

(71 citation statements)

References 32 publications

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“…Let us solve the system of equations (2.62) for the matrices with non-commutative However, at this stage, we do not a priori assume the defining relations of U q (gl(n)) explicitly 14 .…”

Section: Solution Of the Zero-curvature Representationmentioning

confidence: 99%

“…(2.96) 14 However, structures on U q (gl(n)) are incorporated in the above conditions on the matrix elements and the fact that we are solving the zero-curvature relation, which comes from Yang-Baxter relations. The variables of the form A (1) and B (2) (resp.…”

Section: Solution Of the Zero-curvature Representationmentioning

confidence: 99%

“…where this equation is defined on the direct product of three sets X × X × X, and R ij acts non-trivially on the i-th and j-th components. The Yang-Baxter maps attract interest mainly in the context of discrete integrable evolution equations 1 [3,4,5,6,7,8,9,10,11,12,13,14]. However, the Hamiltonian structures for the Yang-Baxter maps were not very clear, and systematic procedures for quantization of them or their applications to Hamiltonian evolution systems did not exist.…”

Section: Introductionmentioning

confidence: 99%

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Quantum groups, Yang–Baxter maps and quasi-determinants

Tsuboi

2018

Nuclear Physics B

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For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators. Thereby obtained a quasi-determinant expression of the quantum Yang-Baxter map associated with the quantum algebra U q (gl(n)). Moreover, the map is identified with products of quasi-Plücker coordinates over a matrix composed of the L-operators. We also consider the quasi-classical limit, where the underlying quantum algebra reduces to a Poisson algebra. The quasi-determinant expression of the quantum Yang-Baxter map reduces to ratios of determinants, which give a new expression of a classical Yang-Baxter map.2 The notation U h (gl(n)) is often used in literatures for this form of presentation of the quantum algebra. The deformation parameter q is related to h as q = e h . We assume that q is generic.3 The normalization commonly used in literatures can be obtained by the replacement:, L −(2) 21 = (L −(1) L −(2) ) 21 (L −(1) L −(2) ) 22 (L −(1) L −(2) ) 23 J 21 J 22 J 23 J 31 J 32 J 33 , L −(2) 31 = (L −(1) L −(2) ) 31 (L −(1) L −(2) ) 32

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“…Let us solve the system of equations (2.62) for the matrices with non-commutative However, at this stage, we do not a priori assume the defining relations of U q (gl(n)) explicitly 14 .…”

Section: Solution Of the Zero-curvature Representationmentioning

confidence: 99%

Section: Solution Of the Zero-curvature Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum groups, Yang–Baxter maps and quasi-determinants

Tsuboi

2018

Nuclear Physics B

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“…Darboux transformations for noncommutative-extended integrable equations were recently constructed; in the case of Grassman-extended NLS equation in [13] and for the supersymmetric KdV equation [27,28,31] and the AKNS system [29]. At the same time, the derivation of noncommutative versions of YB maps has gained its interest [11].…”

Section: Introductionmentioning

confidence: 99%

Grassmann extensions of Yang–Baxter maps

Grahovski¹,

Konstantinou-Rizos²,

Михайлов³

2016

J. Phys. A: Math. Theor.

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In this paper we show that there are explicit Yang-Baxter maps with Darboux-Lax representation between Grassmann extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative Nonlinear Schrödinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the Yang-Baxter property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional Yang-Baxter maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. We consider their vector generalisations.

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“…In this paper, we study maps obeying the Coxeter relations, which are obtained from the non-commutative discrete KP system of equations (called originally in [52] the non-Abelian Hirota-Miwa system). In the commutative case, such maps were studied in [23,34,59], see also [17] for a version with certain commutativity restrictions. We remark that Hirota's discrete KP equation [31] gives as reductions majority of known integrable systems [40], both discrete and continuous.…”

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confidence: 99%

The Coxeter relations and KP map for non-commuting symbols

Doliwa

Noumi

2020

Lett Math Phys

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We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal’cev–Newmann division ring. The action is constructed from the non-Abelian Hirota–Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case, and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.

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Non-Commutative Rational Yang–Baxter Maps

Cited by 30 publications

References 32 publications

Quantum groups, Yang–Baxter maps and quasi-determinants

Quantum groups, Yang–Baxter maps and quasi-determinants

Grassmann extensions of Yang–Baxter maps

The Coxeter relations and KP map for non-commuting symbols

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