Every ribbon graph order has a Nakayama automorphism making it twisted 1-Calabi-Yau, and it is symmetric, hence 1-Calabi-Yau, precisely when the ribbon graph is bipartite or the field has characteristic two.
Non-commutative nodal curves and derived tame algebras
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In this paper, we develop a geometric approach to study derived tame finite dimensional associative algebras, based on the theory of non-commutative nodal curves.
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Calabi-Yau properties of ribbon graph orders
Every ribbon graph order has a Nakayama automorphism making it twisted 1-Calabi-Yau, and it is symmetric, hence 1-Calabi-Yau, precisely when the ribbon graph is bipartite or the field has characteristic two.