Lie-Poisson and Euler-Poincaré Reduction

Abstract

Besides the Poisson structure on a symplectic manifold, the Lie-Poisson bracket on g*, the dual of a Lie algebra, is perhaps the most fundamental example of a Poisson structure. We shall obtain it in the following manner. Given two smooth functions F, H ∈ F(g*), we extend them to functions F _L , H _L (respectively, F _R , H _R ) on all T*G by left (respectively, right) translations. The bracket {F _L , H _L } (respectively, {F _R , H _R }) is taken in the canonical symplectic structure Ω on T*G. The result is then restricted to g* regarded as the cotangent space at the identity; this defines {F, H}. We shall prove that one gets the Lie-Poisson bracket this way. This process is called Lie-Poisson reduction. In §14.6 we show that the symplectic leaves of this bracket are the coadjoint orbits in g*.