William Hautekiet

Partial modules over a Hopf algebra are a generalization of usual modules: the action should no longer be associative, but only partially associative. These were introduced in [parmod] and can be thought of as a linearization and generalization of partial actions of groups.

The category of partial modules is a biactegory over the category of global modules over H: the tensor product of a partial module and a global module is again a partial module. 

Looking at the relative center of this biactegory, we obtain a category of partial Yetter-Drinfel'd modules H_par YD^H, which is monoidal, but not braided. 

We prove a partial module version of a theorem of T. Brzezinski and G. Militaru [BM]: if A is a "commutative'' algebra in H_par YD^H, then the partial smash product A#H has the structure of a Hopf algebroid. 

Joint work with Eliezer Batista (Universidade Federal de Santa Catarina) and Joost Vercruysse (Université Libre de Bruxelles).

References

[parmod] M. M. S. Alves, E. Batista, J. Vercruysse,  Partial representations of Hopf algebras, J. Algebra 426 (2015), 137--187.

[BM] T. Brzezinski, G. Militaru, Bialgebroids, (X_A)-bialgebroids and duality, J.\ Algebra 251 (2022), 279--294.