On the Zero Divisors of Hopf Algebras

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 di...

On the Coderivations of Some Hopf Algebras

A new proof of the Frobenius conjecture on the dimensions of real algebras without zero divisors

A new way to prove the Frobenius conjecture on the dimensions of real algebras without zero divisors is given in the present paper. Firstly, the proof of nonexistence of real algebras without zero divisors in all dimensions except 1,2,4 and 8 was given in [1]. It was based on the simplicial cohomology operation technique. Later on the methods of K-theory cohomology operations gave one a possibi...

Adjunctions between Hom and Tensor as endofunctors of (bi-) module category of comodule algebras over a quasi-Hopf algebra.

For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor endofunctors V k - and - kV are left adjoint to some kinds of  Hom-endofunctors of _HM. The units and counits of these adjunctions are formally trivial as in the classical case.The category of (bi-) modules over a quasi-Hopf algebra is monoidal and some generalized versions of  Hom-tensor relations have been st...

On the cyclic Homology of multiplier Hopf algebras

In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ consisting of a regular multiplier Hopf algebra $mathcal{H}$, a left $mathcal{H}$-comodule algebra $mathcal{R}$, and a unital left $mathcal{H}$-module $mathcal{X}$ which is also a unital algebra. First, we construct a para...