Algebraic and Geometrical Aspects of Rational Gaudin Models

In chapter 2 we give an essential review of the bihamiltonian theory of So V, just illustrating the main theorems. Special emphasis is put on the theory of So V for G Z manifolds admitting an affine structure. We give also a brief survey on Lie-Poisson manifolds and on Poisson product structure with the aim of introducing concepts, notations and conventions to be widely used in subsequent chapters. • In chapter 3 we recall the history of rational Gaudin models and we give a detailed description of the results previously obtained on the set of parameter independent integrals (l.0.11,1.0.12) by Ballesteros and Ragnisco on one side and by Flaschka, Kapovich and Millson on the other. In chapter 4 we introduce a bihamiltonian structure for the inhomogeneous Gaudin model using the results of Reyman and Semenov-Tian-Shansky. vVe show that the Lax and the bihamiltonian approach are essentially equivalent. vVe comment on the differences arising -vvhen the spectrum of the matrix O" in (1.0.6) is not simple. • In chapter 5 we concentrate on the set of parameter independent integrals. VVe introduce a further Poisson tensor R that is compatible with P but not -vvith Q. 'vVe show that the Hamiltonians of the GZ sequences associated with P and R can be seen as the spectral invariants of N Lax matrices. VVe prove that such invariants define a completely integrable Hamiltonian system on PC:N for fl belonging to the class of classical Lie algebras. • In chapter 6 we apply the machinery of bihamiltonian theory of Sov to separate the set of parameter independent Hamiltonians just obtained in chapter 5 for the case fl = sl ( r). We show that the separation coordinates are given by particular points on the spectral curves associated with our Lax matrices. \Ve have N Lax matrices and they have all the same genus g so that in our case the genus of the spectral curves is independent from the number of "particles" N. We give explicit examples of the SoV procedure in the cases fl = sl (2) and fl = sl (3). • In the last chapter we give some preliminary results on the quantization of the integrals we obtained by the GZ sequences associated with the Poisson tensors P and R. • Finally, in appendices we give the proofs of two lemmas that were too cumbersome to be placed in the text.