On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras

Abstract

For a class of neither pointed nor semisimple Hopf algebras H4n of dimension 4n, it is shown that they are quasi-triangular, which universal R-matrices are described. The corresponding weak Hopf algebras wH4n and their representations are constructed. Finally, their duality and their Green rings are established by generators and relations explicitly. It turns out that the Green rings of the associated weak Hopf algebras are not commutative even if the Green rings of H4n are commutative.

On Hopf algebras over basic Hopf algebras of dimension 24

“…It is a 12-dimensional nopointed basic Hopf algebras as the special kind of the only non-pointed basic Hopf algebra of dimension 4p, whose Green ring is determined in [CYWX18]. In particular,…”

Classification of finite-dimensional Hopf algebras over dual Radford algebras

“…The Hopf algebra H p,−1 is the dual of a Radford algebra A p,−1 [28] and its Green ring was determined in [17]. In particular, if p is a prime number, then H p,−1 is the only non-pointed basic Hopf algebra of dimension 4p [9].…”