Hopf-Type Algebras

Abstract

In this article, we provide an alternative approach to the definition of a weak Hopf algebra (WHA). For an associative unital algebra A with a coassociative comultiplication Δ ∈Alg u (A, A ⊗ A), the set of homomorphisms from A to A ⊗ A, which do not preserve the units. If the linear maps Ξ1, Ξ2 ∈ End(A ⊗ A), defined by Ξ1(a ⊗ b) = Δ(a)(1 ⊗ b), Ξ2(a ⊗ b) = (a ⊗ 1)Δ(b), are von Neumann regular elements in the ring End(A ⊗ A) of endomorphisms of A ⊗ A satisfying some appropriate assumptions, we call the A a Hopf-type algebra. We show the existence of a target, a source, a counit, and an antipode of A as in the usual WHA.