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We consider the problem of building non-invertible quantum symmetries (as characterized by actions of unitary fusion categories) on noncommutative tori. We introduce a general method to construct actions of fusion categories on inductive limit C*-algberas using finite dimenionsal data, and then apply it to obtain AT-actions of arbitrary Haagerup-Izumi categories on noncommutative 2-tori, of the even part of the E₈ subfactor on a noncommutative 3-torus, and of PSU (2) ₁₅ on a noncommutative 4-torus.
Evans et al. (Mon,) studied this question.