2006/04/30 by E. Mukhin, E Mukhin, Alexander Varchenko +1 · 3 citations
Mathematics · #Action (physics) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic structures and combinatorial models #Bethe ansatz #Conjecture #Eigenvalues and eigenvectors #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum mechanics #Simple (philosophy) #Tensor product #Trigonometry #math.QA #msc:17B67 #msc:82B23
paper · pdf · doi:10.2140/gtm.2008.13.385
published as Geom. Topol. Monogr. 13 (2008) 385-420 · This is the version published by Geometry & Topology Monographs on 19 March 2008
openalex publication_date 2008/03/19 · arxiv created 2009/04/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract. We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finite-dimensional representations. Having one solution, we describe a construction of new solutions. The collection of all solutions obtained from a given one is called a population. We show that the Weyl group of g acts on the points of a population freely and transitively (under certain conditions). To a solution of the Bethe Ansatz equation, one assigns a common eigenvector (called the Bethe vector) of the trigonometric Gaudin operators. The dynamical Weyl group projectively acts on the common eigenvectors of the trigonometric Gaudin operators. We conjecture that this action preserves the set of the Bethe vectors and coincides with the action induced by the action on points of populations. We prove the conjecture for sl2. 1.