Universal scaling relations for dissipative Tomonaga-Luttinger (TL) liquids with
SU(N) spin symmetry are obtained for both fermions and bosons, by using asymptotic Bethe-ansatz solutions and conformal field theory (CFT) in one-dimensional non-Hermitian quantum many-body systems with
SU(N) symmetry. We uncover that the spectrum of dissipative TL liquids with
SU(N) spin symmetry is described by the sum of one charge mode characterized by a complex generalization of
c=1 U(1) Gaussian CFT, and $N\ensuremath{-}1$ spin modes characterized by level-
1SU(N) Kac-Moody algebra with the conformal anomaly $c=N\ensuremath{-}1$, and thereby dissipation only affects the charge mode as a result of spin-charge separation in one-dimensional non-Hermitian quantum systems. The derivation is based on a complex generalization of Haldane's ideal-gas description, which is implemented by the
SU(N) Calogero-Sutherland model with inverse-square long-range interactions.
A simple set of algebraic equations is derived for the exact low-temperature thermodynamics of one-dimensional multicomponent strongly attractive fermionic atoms with enlarged SU
(N) spin symmetry and Zeeman splitting. Universal multicomponent Tomonaga-Luttinger liquid (TLL) phases are thus determined. For linear Zeeman splitting, the physics of the gapless phase at low temperatures belongs to the universality class of a two-component asymmetric TLL corresponding to spin-neutral
N-atom composites and spin-$(N\ensuremath{-}1)/2$ single atoms. The equation of state which we obtained provides a precise description of multicomponent composite fermions and opens up the study of quantum criticality in one-dimensional systems of
N-component Fermi gases with population imbalance.