Invariants from Group Algebras via Topological Quantum Field Theory
AuthorGrifall-Sabo, Em (John)
Mathematics & Statistics
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We describe a classical characterization of a Frobenius algebra $ A $ as an associative algebra equipped with a comultiplication $ \delta $ which is $ A $-linear. We use this characterization to establish the equivalence of categories between commutative Frobenius algebras and two-dimensional topological quantum field theories, a fact which is well known to experts. We then use the equivalence to derive topological invariants for closed oriented surfaces, such as the genus of a surface, using Frobenius algebras. We use the above results to provide a partial identification of those Frobenius structures on a group algebra which distinguish between closed oriented surfaces of any genus.