Yang–Baxter maps and integrable dynamics

By A. Veselov. 2002

163 citations recorded by the source corpus.

Abstract The hierarchy of commuting maps related to a set-theoretical solution of the quantum Yang–Baxter equation (Yang–Baxter map) is introduced. They can be considered as dynamical analogues of the monodromy and/or transfer-matrices. The general scheme of producing Yang–Baxter maps based on matrix factorisation is discussed in the context of the integrability problem for the corresponding dynamical systems. Some examples of birational Yang–Baxter maps coming from the theory of the periodic dressing chain and matrix KdV equation are discussed.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/79563e3e54e46319af1138bff942944521879199

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In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Geometry of the Yang-Baxter Maps: pencils of conics and quadrirational mappings (2003, methodology): “Important examples of Yang–Baxter maps were found in [1, 12, 10, 6, 15, 8].”
Burchnall-Chaundy Theory for q-Difference Operators and q-Deformed Heisenberg Algebras (2003, methodology): “t x ∈ X, parameter λ and additional (“spectral”) parameter ζ ∈ C, which satisﬁes the following relation: (5) A(x,λ;ζ)A(y,µ;ζ) = A(˜y,µ;ζ)A(˜x,λ;ζ), whenever (˜x, ˜y) = R(λ,µ)(x,y). As it was shown in [2], such a matrix allows one to produce integrals for the dynamics of the related transfer-maps. Ou”