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We use minimal tilting complexes to construct an explicit bijection betweenthe set of thick tensor ideals with the two-out-of-three property in thecategory of finite-dimensional modules over a quantum group at a root of unityand the set of thick tensor ideals in the subcategory of tilting modules. Wealso explain why the analogous construction for rational representations of areductive algebraic group over a field of positive characteristic does not giverise to a bijection.

Tensor ideals for quantum groups via minimal tilting complexes