Uniqueness of higher Gaudin hamiltonians

TL;DR

This paper proves the uniqueness of higher Gaudin Hamiltonians by showing that the quantum algebra is determined by its classical quadratic elements, unifying different construction methods.

Contribution

It establishes that the quantum higher Gaudin Hamiltonians are uniquely determined by the classical quadratic subalgebra, confirming the equivalence of different construction approaches.

Findings

The classical subalgebra is the Poisson centralizer of certain quadratic elements.

The quantum subalgebra is uniquely determined by the classical quadratic elements.

Different methods for constructing higher Gaudin Hamiltonians produce the same operators.

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.