论文标题
分级$ k $ - 理论和莱维特路径代数
Graded $K$-theory and Leavitt path algebras
论文作者
论文摘要
令$ g $为一个组，$ \ ell $ a commutative unital $ \ ast $ ring，带有元素$λ\ in \ ell $ in \ ell $，这样$λ+λ^\ ast = 1 $。我们介绍了$ \ ast $ -Algebras的Hermitian Bivariant $ k $ -k $ - 配备$ G $ ACTION或$ G $ - 级别的变体。对于任何具有有限多个顶点和任何重量功能$ω\ colon e^1 \ to g $的图形$ e $，在$ l（e）= l_ \ ell（e）$ g $ g $ g $ g $ g $ g $ g $ k $ k $ k $ k $ k $ kk^h_ kk^h_ kk^h_ kk^h_ $ l（e）$作为矩阵的圆锥，其系数为$ \ mathbb {z} [g] $与$ e $的发射矩阵和权重$ω$相关的$。在标准$ \ mathbb {z} $的特殊情况下，在$ \ ell $上的轻度假设下，我们表明$ kk^h _ {\ mathbb {z} _ {z} _ {\ nathrm {gr}} $的同构$ l（e）$ in $ kk^h _ {\ mathbb {z} _ {我们还获得了分级和Hermitian分级$ k $的结果 - 总体上，尤其是Leavitt Path代数，尤其是Leavitt Path代数，其中包括Dade Theorem的Hermitian和Bivariant版本的Dade Theorem和van den Bergh的确切级别级别的层次和未级别的$ K $ - $ K $。
Let $G$ be a group and $\ell$ a commutative unital $\ast$-ring with an element $λ\in \ell$ such that $λ+ λ^\ast = 1$. We introduce variants of hermitian bivariant $K$-theory for $\ast$-algebras equipped with a $G$-action or a $G$-grading. For any graph $E$ with finitely many vertices and any weight function $ω\colon E^1 \to G$, a distinguished triangle for $L(E)=L_\ell(E)$ in the hermitian $G$-graded bivariant $K$-theory category $kk^h_{G_{\mathrm{gr}}}$ is obtained, describing $L(E)$ as a cone of a matrix with coefficients in $\mathbb{Z}[G]$ associated to the incidence matrix of $E$ and the weight $ω$. In the particular case of the standard $\mathbb{Z}$-grading, and under mild assumptions on $\ell$, we show that the isomorphism class of $L(E)$ in $kk^h_{\mathbb{Z}_{\mathrm{gr}}}$ is determined by the graded Bowen-Franks module of $E$. We also obtain results for the graded and hermitian graded $K$-theory of $\ast$-algebras in general and Leavitt path algebras in particular which are of independent interest, including hermitian and bivariant versions of Dade's theorem and of Van den Bergh's exact sequence relating graded and ungraded $K$-theory.
