Tensor products of Leavitt path algebras

Ara, Pere; Cortiñas, Guillermo

doi:10.1090/S0002-9939-2013-11561-3

Tensor products of Leavitt path algebras
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Proc. Amer. Math. Soc. 141 (2013), 2629-2639 Request permission

Abstract:

We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular, they are not isomorphic. Similarly, $L_\infty$ and $L_\infty \otimes L_\infty$ are distinguished by their Hochschild homologies, and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_*(L_2)=K_*(L_2\otimes L_2)=0$ and $K_*(L_\infty )=K_*(L_\infty \otimes L_\infty )=K_*(k)$.

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