BPS states for scalar field theories based on ${\mathfrak{g}}_2$ and $\mathfrak{s}\mathfrak{u}(4)$ algebras

We discuss two models in $1+1$ dimensional space-time for real scalar field multiplets on the root space of ${\mathfrak{g}}_2$ and $\mathfrak{s}\mathfrak{u}(4)$ Lie algebras. The construction of these models is presented and the corresponding BPS solutions are found.