Commutator Hopf Subalgebras and Irreducible Representations

Abstract

Montgomery and Witherspoon proved that upper and lower semisolvable, semisimple, finite dimensional Hopf algebras are of Frobenius type when their dimensions are not divisible by the characteristic of the base field. In this note we show that a finite dimensional, semisimple, lower solvable Hopf algebra is always of Frobenius type, in arbitrary characteristic.

Key Words:

• Finite dimensional
• Frobenius type
• Hopf algebra
• Kaplansky conjecture
• Solvable

Mathematics Subject Classification:

• Primary 16W30
• Secondary 16S40

ACKNOWLEDGMENTS

This research was supported in part by a grant from the National Security Agency.

My thanks to R. Guralnick and D. Passman for helpful explanatory comments regarding (2.7). My thanks to the referee for several helpful remarks and in particular for pointing me toward the examples discussed in Section 4.

Notes

Communicated by M. Cohen.