Zero Divisors of Hopf Algebras: A Generalization of the Classical Zero-Divisor Conjecture

In an attempt to study the zero divisors in Hopf algebras, we study four non-trivial examples of nongroup ring infinite Hopf algebras and show that a variant of Kaplansky’s classical zero divisor conjecture holds for these four Hopf algebras. In particular we proof that the Hopf algebra CG has no non-trivial zero-divisors for any infinite finitely generated group G. We then consider the zero divisors of the Hopf algebra of representative functions over compact groups and prove our conjecture for a special case.