Graded bimodules over split group algebras: stabilizers and Peirce decompositions

Abstract: We classify, up to graded isomorphism, graded bimodules over split group algebras of finite abelian subgroups of an arbitrary group. The classification is governed by double cosets, stabilizer characters, and coefficient modules. Over a field, we determine the essential image of the induction functor on Yetter-Drinfel'd modules, giving a negative answer to a question of Dos Santos and Yasumura in the nonabelian case. We also compute the Peirce components and characterize multiplicity-free bimodules through extensions of stabilizer characters.