Uniqueness of higher Gaudin hamiltonians

Abstract

For any semisimple Lie algebra $\mathfrak{g}$, the universal enveloping algebra of the infinite-dimensional pro-nilpotent Lie algebra $\mathfrak{g}_-:=\mathfrak{g}\otimes t^{-1}\mathbb{C}[t^{-1}]$ contains a large commutative subalgebra $\mathcal{A}\subset U(\mathfrak{g}_-)$. This subalgebra comes from the center of the universal enveloping of the affine Kac--Moody algebra $\hat{\mathfrak{g}}$ at the critical level and gives rise to the construction of higher hamiltonians of the Gaudin model (due to Feigin, Frenkel and Reshetikhin). Though there are no explicit formulas for the generators of $\mathcal{A}$ known in general, the "classical analogue" of this subalgebra, i.e. the associated graded subalgebra in the Poisson algebra $A\subset S(\mathfrak{g}_-)$, can be easily described. In this note we show that the "classical" subalgebra $A\subset S(\mathfrak{g}_-)$ is the Poisson centralizer of some of its quadratic elements, and deduce from this that the "quantum" subalgebra $\mathcal{A}\subset U(\mathfrak{g}_-)$ is uniquely determined by the space of quadratic elements of the classical one. In particular, this means that some different constructions of higher Gaudin hamiltonians (namely, Feigin-Frenkel-Reshetikhin's method and Talalaev-Chervov's method), give the same family of commuting operators. The proof uses some ideas of the previous paper math.QA/0608586.