We investigate a metallic zigzag carbon nanotube by means of a Hubbard model, which includes both on-site and nearest-neighbor interactions. Assuming weak interactions, a renormalization group analysis of the equivalent two-leg ladder followed by bosonization and refermionization results in a Gross-Neveu model with an enlarged symmetry relative to the original Hamiltonian. For the undoped case the symmetry of the Gross-Neveu model is SO_8_, but for the doped case the particle-hole symmetry is broken and the symmetry reduces to SO_6_. Four ground-state phases are found in the undoped carbon nanotube with repulsive interactions, a d-wave Mott insulator, an s-wave Mott insulator, a p-density wave, and a charge density wave. The doped case has two ground-state phases, a d-wave superconductor and a phase where a p density wave and a charge density wave coexist. We also explore the global phase diagram with a general interaction profile and find several additional states, including a chiral current phase where current flows around the nanotube along the zigzag bonds.