Symplectic and Poisson structures on some loop groups.(English)Zbl 0820.58021

Maeda, Yoshiaki (ed.) et al., Symplectic geometry and quantization. Papers presented at the 31st Taniguchi international symposium on symplectic geometry and quantization problems held at Sanda, Japan, July 12-17, 1993 and a satellite symposium held at Keio University, Yokohama, Japan, from July 21-24, 1993. Providence, RI: American Mathematical Society. Contemp. Math. 179, 173-192 (1994).
The author’s abstract: Drinfeld has given an idea of multiplicative Poisson structures on Lie groups. The essential derivative of a multiplicative Poisson structure at the unit element of a group defines a Lie algebra structure on the dual space of the Lie algebra of the group. It is known that classical $$r$$-matrices give multiplicative Poisson structures. In this paper we find classical $$r$$-matrices of some loop groups and study dual Lie algebra structures defined by them”.
For the entire collection see [Zbl 0810.00022].

MSC:

 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 22E67 Loop groups and related constructions, group-theoretic treatment 53D50 Geometric quantization