  I discuss mirrors of Landau-Ginzburg models formed by a minimal semisimple
adjoint orbit of $\mathfrak{sl}(n)$ together with a potential obtained via the
Cartan-Killing form. I show that the Landau-Ginzburg models produced by the
Gross-Siebert recipe give precisely the objects of the desired mirrors.
  It is known that Landau-Ginzburg model $LG(2)$ over the semisimple adjoint
orbit of $\mathfrak{sl}(2)$ does not have projective mirrors. I prove
Homological Mirror Symmetry for $LG(2)$ by constructing a Landau-Ginzburg
mirror and showing that its Orlov category of singularities is equivalent to
$Fuk(LG(2))$.
