The injective and projective Leavitt complexes

For a certain finite graph E, we consider the corresponding finite dimensional algebra A with radical square zero. An explicit compact generator for the homotopy category of acyclic complexes of injective (resp. projective) modules over A, called the injective (resp. projective) Leavitt complex of E, was constructed in [18] (resp. [19]). We overview the connection between the injective (resp. projective) Leavitt complex and the Leavitt path algebra of E. A differential graded bimodule structure, which is right quasi-balanced, is endowed to the injective (resp. projective) Leavitt complex in [18] (resp. [19]). We prove that the injective (resp. projective) Leavitt complex is not left quasi-balanced.