The quintessence with Abelian and non-Abelian symmetry

We study the perturbations on both "radial" and "angular" components of the quintessence with an internal Abelian and non-Abelian symmetry. The properties of the perturbation on the "radial" component depend on the specific potential of the model and is similiar for both Abelian and non-Abelian case. We show that the cosine-type potential is very interesting for the O(N) quintessence model and also give a critical condition of instability for the potential. While the properties of perturbations on "angular" components depend on whether the internal symmetry is Abelian or non-Abelian, which we have discussed respectively. In the non-Abelian case, the fluctuation of the "angular" component will increase rapidly with time while in the Abelian case it will not.