Abstract
The main work of this article is to give a construction of semilattice graded weak Hopf algebras via S-Hopf quivers, mainly based on the work of Cibils, Rosso, and Montgomery. This provides a class of non-commutative and non-cocommutative pointed semilattice graded weak Hopf algebras: they have the natural bases consisting of paths and the underlying coalgebra structures are path coalgebras. It also provides a new way of verifying old results and testing new ideas on pointed Hopf algebras, such as the decomposition of pointed semilattice graded weak Hopf algebras according to the result of Montgomery.
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