The Leavitt path algebra of a graph

Abstract For any row-finite graph E and any field K we construct the Leavitt path algebra L ( E ) having coefficients in K . When K is the field of complex numbers, then L ( E ) is the algebraic analog of the Cuntz–Krieger algebra C ∗ ( E ) described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix rings M n ( K ) and the Leavitt algebras L ( 1 , n ) appear as algebras of the form L ( E ) for various graphs E . In our main result, we give necessary and sufficient conditions on E which imply that L ( E ) is simple.