The injective and projective Leavitt complexes

Let A be a finite dimensional algebra over a field k. We denote by Kac(A-Inj) the homotopy category of acyclic complexes of injective A-modules which are called the stable derived category of A in [16]. This category is a compactly generated triangulated category such that its subcategory of compact objects is triangle equivalent to the singularity category [4, 22] of A. In general, it seems very difficult to give an explicit compact generator for the stable derived category of an algebra. An explicit compact generator called the injective Leavitt complex, for the homotopy category Kac(A-Inj) in the case that the algebra A is with radical square zero was constructed in [18]. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex is quasi-isomorphic to the Leavitt path algebra in the sense of [2, 3]. Here, the Leavitt path algebra is naturally Z-graded and viewed as a differential graded algebra with trivial differential. We denote byKac(A-Proj) the homotopy category of acyclic complexes of projective A-modules. This category is a compactly generated triangulated category whose subcategory of compact objects is triangle equivalent to the opposite category of the singularity category of the opposite algebra Aop. An explicit compact generator, called the projective Leavitt complex, for the homotopy category Kac(A-Proj) in the case that A is an algebra with radical square zero was constructed in [19]. It is shown that the opposite differential graded endomorphism algebra of the projective Leavitt complex of a finite quiver without sources is quasi-isomorphic to the Leavitt path algebra of the opposite graph [19]. We recall the constructions of the injective and projective Leavitt complexes. We overview the connection between the injective and projective Leavitt complexes and the Leavitt path algebras of the given graphs. A differential graded bimodule structure, which is right quasi-balanced, is endowed to the injective and projective Leavitt complex in [18, 19]. We prove that neither the injective nor the projective Leavitt complex is not left quasi-balanced.