A commuting pair in Hopf algebras

Abstract

Working over an algebraically closed field of characteristiczero, it is shown that for a finite dimensional semisimple Hopfalgebra H, the actions of the Drinfeld double D(H) and thecharacter algebra C(H)\subset H\sp * form a commuting pair in{\rm End}(H). A result of G. I. Kats says that the trace ofright multiplication on H\sp * by any minimal idempotent ofC(H) is a divisor of \dim(H). These results enable the authorto prove that the dimension of every simple D(H) submodule inH is a divisor of \dim(H)