The Graded Hopf Algebra Structure Of Finite Dimensional Leavitt Path Algebras

Firstly,it is proved that the finite dimensional Leavitt path algebra has the structure of Z-graded bialgebra if and only if its underlying graph has isolated vertices.Secondly,on the basis of the above conclusion,by defining suitable antipodes,it is proved that the Leavitt path algebra on a directed graph each of whose path has length no more than 1 forms a Z-graded Hopf algebra if and only if the underlying graph contains only isolated vertices.In addition,we prove that the Leavitt path algebras on a directed graph with only one path of length 2 and on a directed graph with one path of length 1 and one path of length 2 can not form a Z-graded Hopf algebra.