Abstract: A quiver is an oriented graph consisting of vertices and arrows. One can construct the path algebras and their quotients of the quivers. This provides finite dimensional elementary algebras in an exhaustive way, due to a well-known theorem of Gabriel. There is a dual analogue for coalgebras given by Chin and Montgomery. A Hopf algebra is simultaneously an algebra and a coalgebra in a compatible way. It is natural to construct Hopf algebras via quivers. We give a survey on this direction and propose to classify finite dimensional pointed (resp. elementary) Hopf algebras via quivers.
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