A bi-Hamiltonian nature of the Gaudin algebras

Abstract

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