Algebras and Representation Theory

, Volume 21, Issue 4, pp 833–858 | Cite as

The Injective Leavitt Complex

  • Huanhuan LiEmail author


For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally \(\mathbb {Z}\)-graded and viewed as a differential graded algebra with trivial differential.


Injective Leavitt complex Compact generator Leavitt path algebra Dg quasi-balanced module 

Mathematics Subject Classification (2010)

16G20 16E45 18E30 18G35 


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The author thanks her supervisor Professor Xiao-Wu Chen for inspiring discussions and encouragement. This project was supported by the National Natural Science Foundation of China (No.s 11522113 and 11571329). The author also gratefully acknowledges the support of Australian Research Council grant DP160101481.


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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Centre for Research in MathematicsWestern Sydney UniversitySydneyAustralia

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