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We describe the Fukaya-Seidel category of a Landau-Ginzburg model LG(2) forthe semisimple adjoint orbit of sl(2,C). We prove that this category isequivalent to a full triangulated subcategory of the category of coherentsheaves on the second Hirzebruch surface. We show that no projective varietycan be mirror to LG(2), and that this remains so after compactification.

A Lie theoretical construction of a Landau-Ginzburg model without projective mirrors