Braided Frobenius algebras

Abstract

Monoidal categories have proven to be especially useful in the analysis of both algebraic structures such as associative algebras and geometric structures such as knots and braids. In this paper, we consider Frobenius algebras. These are algebraic structures consisting of an associative algebra and a coassociative coalgebra, satisfying a compatibility relation. Frobenius algebras have many applications in algebra and computer science. They have also been shown to characterize low-dimensional topological quantum field theories. They are traditionally considered in symmetric monoidal categories. But we generalize the theory to braided monoidal categories. In the process, we obtain a number of new examples of this algebraic notion. Our examples arise in categories of crossed G-sets, categories of representations of quasitriangular Hopf algebras and several geometric categories.