Tensor products of Leavitt path algebras

TL;DR

This paper calculates the Hochschild homology of Leavitt path algebras, demonstrating that certain tensor products are not Morita equivalent despite having identical K-theory, thus providing new algebraic invariants.

Contribution

It computes Hochschild homology for Leavitt path algebras and shows these invariants distinguish tensor products that K-theory cannot.

Findings

Hochschild homology differentiates $L_2$ and $L_2\otimes L_2$

Hochschild homology distinguishes $L_\infty$ and $L_\infty\otimes L_\infty$

K-theory is insufficient to distinguish these algebras

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 $K_*(L_\infty)=K_*(L_\infty\otimes L_\infty)=K_*(k)$.