A bi-Hamiltonian nature of the Gaudin algebras

Let q be a Lie algebra over field k and p,p˜∈k[t] two different normalised polynomials of degree n⩾2. As vector spaces, both quotient algebras q[t]/(p) q[t]/(p˜) can identified with W=q⋅1⊕qt¯⊕…⊕qt¯n−1. If deg⁡(p−p˜)⩽1, then the brackets [,]p, [,]p˜ induced on W by p p˜, respectively, are compatible. Making use Lenard–Magri scheme, we construct subalgebra Z=Z(p,p˜)⊂S(W)q⋅1 such that {Z,Z}p={Z,Z}p˜=0. tr.degS(q)q=indq has codim–2 property, tr.degZ takes maximal possible value, which is n−12dim⁡q+n+12indq. q=g semisimple, Z contains Hamiltonians suitably chosen Gaudin model. Furthermore, if p˜ do not have common roots, there C⊂U(g⊕n) Z=gr(C), up to certain identification. In non-reductive case, obtain completely integrable generalisation models. For wide class algebras, extends reductive setting, Z(p,p+t) coincides image Poisson-commutative Z(qˆ,t)=S(tq[t])q[t−1] under map ψp:S(q[t])→S(W), providing p(0)≠0.