Two-dimensional Topological Quantum Field Theories and Frobenius Algebras
نویسنده
چکیده
We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either \annihilator algebras" | algebras whose socle is a principal ideal | or eld extensions. The relationship between two-dimensional topological quantum eld theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classiication of the indecomposable two-dimensional topological quantum eld theories.
منابع مشابه
2-dimensional Topological Quantum Field Theories and Frobenius Algebras
Category theory provides a more abstract and thus more general setting for considering the structure of mathematical objects. 2-dimensional quantum field theories arise in physics as objects that assign vector spaces to 1-manifolds and linear maps to 2-cobordisms. From a categorical perspective, we find that they are the same as commutative Frobenius algebras. Our main goal is to explain this e...
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