Relative cellular algebras

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are relative cellular, but not cellular. Most prominently, the restricted enveloping algebra and the small quantum group for \(\mathfrak{sl}_{2}\), and an annular version of arc algebras.