The Concept of Parafree Zinbiel Algebras

Pf (parafree) Zinbiel (PfZin) algebras, a generalization of Leibniz algebras, share various traits with free Zinbiel algebras. This article delves into the intricacies of PfZin algebras, presenting their structure and exploring significant findings analogous to those in parafree Leibniz algebras. The focus extends to properties of subalgebras and quotient algebras within the realm of PfZin algebras. Additionally, the direct sum of these algebras is examined, demonstrating that the amalgamation of two PfZin algebras yields a Zinbiel algebra. A new connection between weak Hopf algebras and PfZin algeras constructed. Moreover, from the direct sum of PfZin algebras weak Hopf algebra is handled and construction of weak Hopf algebra usuing PfZin algebra is showed.