## Two-Dimensional Topological Quantum Field Theories And Frobenius Algebras (1996)

Venue: | J. Knot Theory Ramifications |

Citations: | 60 - 2 self |

### BibTeX

@ARTICLE{Abrams96two-dimensionaltopological,

author = {Lowell Abrams},

title = {Two-Dimensional Topological Quantum Field Theories And Frobenius Algebras},

journal = {J. Knot Theory Ramifications},

year = {1996},

volume = {5}

}

### OpenURL

### Abstract

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 field extensions. The relationship between two-dimensional topological quantum field 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 classification of the indecomposable two-dimensional topological quantum field theories. Keywords: topological quantum field theory, frobenius algebra, two-dimensional cobordism, category theory 1. Introduction Topological Quantum Field Theories (TQFT's) were first described axiomatically by Atiyah in [1]. Since then, much work has been done ...

### Citations

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Citation Context ...t's account.) Nevertheless, there has been difficulty formulating this theorem precisely and filling in the details of its proof [9, 10]. Indeed, the existing literature on the structure of FA's (see =-=[11]-=- and [12]) does not seem sufficient to support such a theorem. In particular, a precise definition of the coalgebra structure of a FA, and an understanding of its relation to the multiplicative struct... |

415 | Linear representations of finite groups - Serre - 1977 |

129 |
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Citation Context ...l FA structure. The next example is similar to cohomology; it includes the case of a truncated polynomial algebra. Example 3. Finite dimensional (graded) commutative connected Hopf algebras. Margolis =-=[18]-=- shows that if A is such an algebra then it is a Poincar'e algebra, i.e. there is an isomorphism A q = An\Gammaq , where n = (A : K). It follows that A is an annihilator algebra with socle generated b... |

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Citation Context ... Frobenius algebra" (FA), and sketches a proof. (See [8] for a physicist's account.) Nevertheless, there has been difficulty formulating this theorem precisely and filling in the details of its p=-=roof [9, 10]-=-. Indeed, the existing literature on the structure of FA's (see [11] and [12]) does not seem sufficient to support such a theorem. In particular, a precise definition of the coalgebra structure of a F... |

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Citation Context ...nt.) Nevertheless, there has been difficulty formulating this theorem precisely and filling in the details of its proof [9, 10]. Indeed, the existing literature on the structure of FA's (see [11] and =-=[12]-=-) does not seem sufficient to support such a theorem. In particular, a precise definition of the coalgebra structure of a FA, and an understanding of its relation to the multiplicative structure, has ... |