We investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras $L(E)$ and $L(F)$ of graphs $E$ and $F$ over a commutative ground ring $\ell$. We approach this by studying the structure of such algebras under bivariant algebraic $K$-theory $kk$, which is the universal homology theory with the properties above. We show that under very mild assumptions on $\ell$, for a graph $E$ with finitely many vertices and reduced incidence matrix $A_E$, the structure of $L(E)$ in $kk$ depends only on the groups Coker$(I-A_E)$ and Coker$(I-A_E^t)$. We also prove that for Leavitt path algebras, $kk$ has several properties similar to those that Kasparov's bivariant $K$-theory has for $C^*$-graph algebras, including analogues of the Universal coefficient and Künneth theorems of Rosenberg and Schochet.

中文翻译：

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代数双变量$ K $-理论和Leavitt路径代数。

我们研究同构不变，激发和矩阵稳定的同源理论在多大程度上可交换图环上的图$ E $和$ F $的Leavitt路径代数$ L（E）$和$ L（F）$ \ ell $。我们通过研究双变量代数$ K $-理论$ kk $下的此类代数的结构来解决这一问题，这是具有上述性质的通用同源性理论。我们显示在$ \ ell $的非常温和的假设下，对于具有有限多个顶点和减少的发生矩阵$ A_E $的图$ E $，$ kk $中的L（E）$的结构仅取决于组Coker $（I-A_E）$和Coker $（I-A_E ^ t）$。我们还证明，对于Leavitt路径代数，$ kk $具有与Kasparov的双变量$ K $-理论对于$ C ^ * $-图代数相似的一些性质，