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We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated with triangulated surfaces with arbitrarily oriented triangles, investigated recently in Erdmann and Skowroński (J Algebra 505:490–558, 2018, Algebras of generalized dihedral type, Preprint. arXiv:1706.00688, 2017). Moreover, we prove that Brauer graph algebras are idempotent algebras of periodic weighted surface algebras, investigated in Erdmann and Skowroński (Algebras of generalized quaternion type, Preprint. arXiv:1710.09640, 2017).

From Brauer graph algebras to biserial weighted surface algebras