Finite presentation of finite tensor categories by finitely many matrices

Determine whether finite tensor categories can be presented by a finite set of matrices, analogous to the finite matrix presentations used for fusion categories.

Background

Non-semisimple Hopf algebras and their representation categories (finite tensor categories) can have continuous families of indecomposable modules, making finite presentations by matrices potentially infeasible compared to the semisimple fusion category case. The paper highlights this difficulty with the example of the 8-dimensional Nichols Hopf algebra K_2, where indecomposable modules include a continuous family parameterized by the 2-sphere.

The authors propose focusing on the projective subcategory as a complete invariant of a finite tensor category to make such a presentation more manageable, and mention plans to show finite presentations in future work; however, the fundamental question of whether a general finite tensor category admits a finite matrix presentation is explicitly stated as unclear in the text.