Juan Cuadra

In [1] and [2], we found an arithmetic difference between group algebras and semisimple Hopf algebras in connection with Kaplansky's sixth conjecture. Namely, there are complex semisimple Hopf algebras that do not admit an integral Hopf order. We provided more instances of this phenomenon in [3]. 

All the examples studied are twists of group algebras in Movshev's way. 

In this talk, we will bring a new perspective on this topic. 

We will present a group-theoretical condition under which a twist of a group algebra admits an integral Hopf order.

The results we will expound are part of the work [4] in collaboration with Ehud Meir (University of Aberdeen, United Kingdom). 

References

[1] J. Cuadra and E. Meir, On the existence of orders in semisimple Hopf algebras. Trans. Amer. Math. Soc. 368 (2016), 2547-2562. 

[2] J. Cuadra and E. Meir, Non-existence of Hopf orders for a twist of the alternating and symmetric groups. J. London Math. Soc. (2) 100 (2019), 137-158.

[3] G. Carnovale, J. Cuadra, and E. Masut, Non-existence of integral Hopf orders for twists of several simple groups of Lie type. Publ. Mat.68 (2024), 73-101. 

[4] J. Cuadra and E. Meir, Existence of integral Hopf orders in twists of group algebras. ArXiv:2211.00097.