Kashina’s exponent conjecture

Formulated by Kashina and recorded by Etingof–Gelaki in 1998, the conjecture asserts that the exponent divides the dimension for every finite-dimensional semisimple and cosemisimple Hopf algebra.

Known results

• Kashina verified several special cases (before 1998).
• Etingof–Gelaki, 1998: the general bound $\exp(H)\mid\dim(H)^3$.
• Etingof–Gelaki, 1998: $[\exp(H)\mid\dim(H)$ for semisimple triangular Hopf algebras in characteristic $0$.

2026 group-theoretical case

A 2026 arXiv paper proves $[\exp(H)\mid\dim_k(H)$ whenever $\operatorname{Rep}(H\otimes_k\overline{k})$ is group-theoretical. It includes prime-power dimensions, dimensions of forms $pq$, $pq^2$, $p^2q$, and $pqr$, and every dimension below $36$; it does not claim the unrestricted conjecture.

Current status (as of August 2026): The conjecture is proved for geometrically group-theoretical Hopf algebras and several dimension classes, while the full finite-dimensional semisimple and cosemisimple case remains open.