Reconstructing algebra objects from NIM-reps in pointed, near-group and quantum group-like fusion categories

Abstract

In this article we study the possible Morita equivalence classes of algebras in three families of fusion categories (pointed, near-group and $A \left( 1,l \right)_{\frac{1}{2}}$) by studying the Non-negative Integer Matrix representations (NIM-reps) of their underlying fusion ring, and compare these results with existing classification results of algebra objects. Also, in an appendix we include a test of the exponents conjecture for modular tensor categories of rank up to 4.