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Bethe algebra and the algebra of functions on the space of differential operators of order two with polynomial kernel. (English) Zbl 1171.82008

The authors consider two algebras \(A_L\) and \(A_G\). The algebra \(A_G\) is the algebra of functions on the intersection of Schubert cycles \(C_{z_1,\Lambda^{(1)}},\dots,C_{z_1,\Lambda^{(n)}},C_{\infty,\Lambda^{(\infty)}}\) in the Grassmannian of two-dimensional subspaces of \(C_d[x].\) The algebra \(A_L\) is generated by the Gaudin Hamiltonians acting on the subspace \(\mathrm{Sing}L_\Lambda [\Lambda^{(\infty)}]\) of singular vectors of weight \(\Lambda^{(\infty)}\) in the tensor product \(L_{\pmb \Lambda}=L_{\Lambda^{(1)}}\otimes\cdots\otimes L_{\Lambda^{(n)}}\) of polynomial irreducible finite-dimensional \(gl_N-\) modules with highest weights \(\Lambda^{(1)},\cdots,\Lambda^{(n)}.\) The main result of this paper is the construction of an isomorphism of \(A_L\) and \(A_G\) for \(N=2.\)

MSC:

82B23 Exactly solvable models; Bethe ansatz
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
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