We 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. We show that the Landau–Ginzburg models produced by the Gross–Siebert recipe give precisely the objects of the desired mirrors. It is known that the Landau–Ginzburg model \(\textrm{LG}\hspace{0.55542pt}(2)\) over the semisimple adjoint orbit of \(\mathfrak {sl}(2)\) does not have projective mirrors. We prove Homological Mirror Symmetry for \(\textrm{LG}\hspace{0.55542pt}(2)\) by constructing a Landau–Ginzburg mirror and showing that its Orlov category of singularities is equivalent to \(\textrm{Fuk}\hspace{0.55542pt}(\textrm{LG}\hspace{0.55542pt}(2))\) .
