Orbit and stabilizer classification for pointed Hopf algebras

Develop an analogue of the orbit and stabilizer classification for graded bimodules over suitable pointed Hopf algebras, identifying the data that replace double cosets, stabilizer characters, and coefficient modules and determining hypotheses under which an explicit Peirce decomposition can be recovered without assuming semisimplicity.

Background

The paper classifies graded bimodules over split group algebras using the orbits of the action of the enveloping group on the grading group, with double cosets indexing orbits, stabilizer characters indexing weights, and arbitrary coefficient modules recording multiplicities. It also derives an explicit Peirce decomposition from this classification without imposing semisimplicity on the coefficient ring or group algebras.

The first further question asks whether this structure persists for suitable pointed Hopf algebras when group gradings are replaced by compatible Hopf coactions. The unresolved issues include identifying the appropriate replacements for double cosets and stabilizers and handling the non-group-like elements of pointed Hopf algebras without relying on semisimplicity.

References

For suitable pointed Hopf algebras in place of $ and $, with compatible Hopf coactions replacing group gradings, is there an analogue of the orbit and stabilizer classification of \cref{thm:classification}? In particular, what data replace double cosets, stabilizer characters, and coefficient modules, and under which hypotheses can one recover an explicit Peirce decomposition?

— Graded bimodules over split group algebras: stabilizers and Peirce decompositions  (2610.06072 - Hindi et al., 5 Oct 2026) in Section 7, Further questions, item (1)

Using the smash-product framework in , can the classification and Peirce formulas be expressed in terms of smash products? The relationship between graded modules and smash-product modules is known. Can the double-coset types, their coefficient modules, and the multiplicity-free compatibility conditions be recognized intrinsically in a smash-product model, and can this perspective clarify composition of graded bimodules?

— Graded bimodules over split group algebras: stabilizers and Peirce decompositions  (2610.06072 - Hindi et al., 5 Oct 2026) in Section 7, Further questions, item (2)

The relative tensor product of two atomic bimodules is computed in Section~7.2. Can this formula be extended to arbitrary coefficient modules and expressed through these equivalences compatibly with associativity when varying the intermediate subgroup? Can the Peirce and multiplicity-free conditions be expressed categorically without choosing representatives or characters?

— Graded bimodules over split group algebras: stabilizers and Peirce decompositions  (2610.06072 - Hindi et al., 5 Oct 2026) in Section 7, Further questions, item (3)