Correspondences of ribbon categories
In this lengthy paper, algebras and their modules internal to a braided tensor category \(\mathcal{C}\) are studied in depth. Most of the theory depends vitally on the braiding not being a symmetry. The left and right centres of a Frobenius algebra \(A\) in \(\mathcal{C}\) are defined and found to have equivalent categories of dyslectic modules in the sense of \textit{B. Pareigis} [``On braiding and dyslexia'', J. Algebra 171, 413--425 (1995; Zbl 0816.18003)]. This is applied to the classification of modular invariants and the construction of two-dimensional conformal field theories. Another result proved here is that, for each modular category \(\mathcal{C}\), there exists a commutative Frobenius algebra \(A\) in a certain semidirect product of \(\mathcal{C}\) and \(\mathcal{C}^{\mathrm{op}}\) such that the category of dyslectic \(A\)-modules is equivalent to the category of vector spaces. This is applied to the study of ``correspondences'' of braided tensor categories.
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