Realization for tensor products of Leavitt path algebras (Q2161399)

This work introduces the so-called augmentation graph associated to two given graphs. It is proved that there is an injective algebra homomorphism from the Leavitt path algebra of the augmentation graph to the tensor product of the Leavitt path algebras of the given graphs. The paper also contains a necessary and sufficient condition on the surjectivity of the homomorphism. This gives rise to a realization of the tensor product of two Leavitt path algebras as a Leavitt path algebra. The authors apply the tool of augmentation graph to approach the following question about the tensor products for Leavitt path algebras which is currently unsolved: Does there exist a (necessarily injective) non-zero ring homomorphism \(\Psi\colon\mathcal{L}_2\otimes\mathcal{L}_2\to \mathcal{L}_2\)? In the work under review, it is shown that there is an injective non-zero ring homomorphism \(\Phi\colon \mathcal{L}_2\to \mathcal{L}_2 \otimes \mathcal{L}_2\).