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Entwining Yang-Baxter maps and integrable lattices

Theodoros E. Kouloukas, Vassilios G. Papageorgiou (2011)

Banach Center Publications

Yang-Baxter (YB) map systems (or set-theoretic analogs of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L₁, L₂, L₃ derived from symplectic leaves of 2 × 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L₁ = L₂ = L₃ this property yields the set-theoretic quantum Yang-Baxter equation, i.e. the...

Gaudin's model and the generating function of the Wroński map

Inna Scherbak (2003)

Banach Center Publications

We consider the Gaudin model associated to a point z ∈ ℂⁿ with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional sl₂-representations, [G]. The Bethe equations of this model provide the critical point system of a remarkable rational symmetric function. Any critical orbit determines a common eigenvector of the Gaudin hamiltonians called a Bethe vector. In [ReV], it was shown that for generic...

Miura opers and critical points of master functions

Evgeny Mukhin, Alexander Varchenko (2005)

Open Mathematics

Critical points of a master function associated to a simple Lie algebra 𝔤 come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra t 𝔤 . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms...

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