the sense explained above. Other examples Let be any semisimple Lie group. Choose a maximal torus and a choice of positive roots. Let be the corresponding opposite Borel subgroups, so that and there is a natural projection . Then define a Lie group which is a subgroup of the product , and has the same dimension as . The standard Poisson–Lie group structure on is determined by identifying the Lie algebra of with the dual of the Lie algebra of , as in the standard Lie bialgebra example. This defines a Poisson–Lie group structure on both and on the dual Poisson Lie group . This is the "standard" example: the Drinfeld-Jimbo quantum group is a quantization of the Poisson algebra of functions on the group . Note that is solvable, whereas is semisimple. See also Lie bialgebra Quantum group Affine quantum group Quantum affine algebras References Lie groups Symplectic geometry Structures on manifolds