The representations of some Hopf algebras have curious behavior: Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. We explain a family of examples of such Hopf algebras and their modules, and classify left, right, and two-sided ideals in their stable module categories.
Benson, D., & Witherspoon, S. (2014). Examples of support varieties for Hopf algebras with noncommutative tensor products. Archiv der Mathematik, 102(6), 513-520. https://doi.org/10.1007/s00013-014-0659-8