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Motivated by recently discovered quasi-one-dimensional superconductorK$_{2}$Cr$_{3}$As$_{3}$ with $D_{3h}$ lattice symmetry, we studyone-dimensional three-orbital Hubbard model with generic electron repulsiveinteraction described by intra-orbital repulsion $U$, inter-orbital repulsion,and Hund's coupling $J$. As extracted from density functional theorycalculation, two of the three atomic orbitals are degenerate ($E^{\prime}$states) and the third one is non-degenerate ($A^{\prime}_1$), and the system ispresumed to be at an incommensurate filling. With the help of bosonization, wehave usual three-band Tomonaga-Luttinger liquid for the normal state. Possiblecharge density wave (CDW), spin density wave (SDW) and superconducting (SC)instabilities are analyzed by renormalization group. The ground state dependson the ratio $J/U$ and is sensitive to the degeneracy of $E^{\prime}$ bands. At$0<J<U/3$, spin-singlet SC state is favored, while spin-tripletsuperconductivity will be favored in the region $U/3<J<U/2$. The SDW state hasthe lowest energy only in the unphysical parameter region $J>U/2$. When thetwo-fold degeneracy of $E^{\prime}$ bands is lifted, SDW instability has thetendency to dominate over the spin-singlet SC state at $0<J<U/3$, while theorder parameter of the spin-triplet SC state will be modulated by a phasefactor $2\Delta k_F x$ at $U/3<J<U/2$. Possible experimental consequences andapplications to K$_{2}$Cr$_{3}$As$_{3}$ are discussed.

Instability of three-band Tomonaga-Luttinger liquid: renormalization group analysis and possible application to K2Cr3As3