Quantum Symmetries of Noncommutative Tori

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 $$E_{8}$$ E 8 subfactor on a noncommutative 3-torus, and of $$\text {PSU}(2)_{15}$$ PSU ( 2 ) 15 on a noncommutative 4-torus.