For now, this is intended primarily as a blackboard-type space for discussions, but could just as well function as a more extensive note-taking and paper-writing space at the same time.
We want to consider Hopf algebras with a Haar integral and a Haar functional, as summarized in Section 1 of Balsam and Kirillov. The conjecture is that for a field of characteristic zero, these Hopf algebras are precisely the algebras of the PROP introduced by Duncan and Dunne in the case of trivial phase group.
Also, how about the PROP for weak Hopf algebras with Haar stuff?
Literature on the Frobenius algebra structure of Hopf algebras
The Larson-Sweedler theorem shows that a Hopf algebra is semisimple if and only if it possesses a left or right integral, which is then unique. It was already known to Larson and Sweedler (remark on p.85) that this equips the Hopf algebra with a Frobenius algebra structure. This was subsequently picked up by Pareigis, who generalized the Larson-Sweedler theorem to Hopf algebras over more general base rings and chose the title of his paper accordingly as “When Hopf algebras are Frobenius algebras”. Other references, in no particular order:
- lecture notes by Farnsteiner (Corollary 1),
- Kadison-Stolin’s Approach to Hopf Algebras via Frobenius Coordinates, parts I and II (not sure what this is about).
- Schauenburg’s Weak Hopf algebras and quantum groupoids presupposes the Frobenius structure implicitly.
- Lomp’s Integrals in Hopf algebras over rings mentions Frobenius in passing.
- Cohen and Fischman do something with Hopf and Frobenius algebras.
- Beattiea, Dăscălescub and Raianub study co-Frobenius Hopf algebras.
- Böhm generalizes some Frobenius things to Hopf algebroids.
- Pareigis continues his study of Frobenius-Hopf algebras and investigates their cohomology.
- Lorenz presents an overview of Frobenius structures in Hopf algebra theory.
- Many more…
In conclusion, the Frobenius algebra structure of a Hopf algebra with Haar integral is well-known. However, the actual PROP for finite-dimensional semisimple Hopf algebras does not seem to have been worked out yet.