Bethe algebra of Gaudin model, Calogero-Moser space, and Cherednik algebra (Q2878703): Difference between revisions

The Bethe algebra of the Gaudin model associated to the complex Lie algebra \(\mathfrak{gl}_N\) of all \(N\times N\) matrices is a commutative subalgebra of the universal enveloping algebra of the current algebra of \(\mathfrak{gl}_N\). The Bethe algebra acts on a subspace \(M\) of any \(\mathfrak{gl}_N[t]\)-module consisting of all vectors of a fixed \(\mathfrak{gl}_N\)weight, producing a commutative family of linear operators \(\mathcal{B}(M)\in \mathrm{End} M\). The authors proved that \(\mathcal{B}(M)\) is naturally isomorphic to the center of the rational Cherednik algebra at the critical level of type \(A\) and that \(\mathcal{B}(M)\) is naturally isomorphic to the algebra \(\mathcal{O}_\chi\) of regular functions on the Calogero-Mozer space.