Compatible Poisson-Lie structures on the loop group of SL2.

Compatible Poisson-Lie structures on the loop group of SL 2. Abstract. We define a 1-parameter family of r-matrices on the loop algebra of sl 2 , defining compatible Poisson structures on the associated loop group, which degenerate into the rational and trigonometric structures, and study the Manin triples associated to them. Introduction. The concept of bi-(or multi-)Hamiltonian structures on manifolds is known, since the works of Magri and Gelfand-Dorfman ([Ma], [GD]), to play an important role in the study of classical integrable systems. Up to now, it still remains unclear whether this notion has any reasonable quantum analogue. Among long-known examples of such structures are the KdV hierarchy phase spaces. In that case, higher compatible Poisson structures are nonlocal, and are expected to be quantized using (nonlocal) vertex operator algebras. This program is very far from its goals at the moment. An other example of bi-Hamiltonian manifolds comes from the theory of Poisson-Lie groups. It is an elementary fact that the rational and trigonometric structures (according to the terminology of [BD]) on loop groups, are compatible. Accordingly, it is a natural question to find higher compatible Poisson structures on these groups. In this paper, we answer this question in the sl 2 case. We perform an elementary manipulation on the rational r-matrix, that after the operation of shifting of the spectral parameter generates the rational and trigonometric r-matrices. This shows the compatibility of these Poisson structures. We then study the associated Manin triples. They are of a slightly different kind from those studied in [D]. It would be interesting to achieve explicitly the quantization of the structures found here. A possible connection between the two problems mentioned above could be the remark, that the Poisson brackets of the KdV monodromy operators in the first (resp. second) Hamiltonian structures are respectively given by the rational and trigonometric r-matrices ([FT]).