  In this work we introduce the concept of Cartan-Polyakov loops, a special
subset of Polyakov loops in the fundamental representation of the
$\mathrm{SU}(N_c)$ group, with charges $k=1,\ldots,(N_c-1)/2$. It constitutes a
sufficient set of independent degrees of freedom and it is used to parametrize
the thermal Wilson line. Polyakov loops not contained in this set are
classified as non-Cartan-Polyakov loops. Using properties of the characteristic
polynomial of the thermal Wilson line, we write a non-Cartan-Polyakov loop
charge decomposition formula. This formalism allows one to readily build
effective models of quarks and gluons with an arbitrary number of colors. We
apply it to the Polyakov$-$Nambu$-$Jona-Lasinio model and to an effective glue
model, in the mean field approximation, showing how to directly extend these
models to higher values of $N_c$.
