Relative cellular algebras

Abstract

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.

arXiv comment39 pages, many figures, revised version, to appear in Transform. Groups, comments welcome

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• Cellularity and subdivision of KLR and weighted KLRW algebras
• Monoidal categories, representation gap and cryptography
• Sandwich cellularity and a version of cell theory
• Cellularity of KLR and weighted KLRW algebras via crystals
• Deforming algebras with anti-involution via twisted associativity