论文标题
交叉产品Leavitt路径代数
Crossed product Leavitt path algebras
论文作者
论文摘要
如果$ e $是指向的图形，而$ k $是一个字段，那么$ e $ $ e $ o $ k $的leavitt路径$ e $ of $ k $上的$ \ mathbb z。我们说明了这一主要结果，以便还以$ L_K（E）$的代数属性来表征$ l_k（e）$的grothendieck $ \ mathbb z $ -group的预购组属性。如果$ e $有限的顶点有限，那么我们表征$ l_k（e）$在$ k_0^γ（l_k（e））的属性方面得到强烈评分。$我们的证明也提供了替代已知的等价证明$ l_k（e）$的替代证明，并且只有在$ e $ n nif的情况下，如果$ e $ nime $ n. $ l_k（e）$在且仅当$ l_k（e）$是交叉产品时，对单位进行了强烈分级和分级。 在显示主要结果的过程中，我们获得了组$γ$的条件和$γ$的分区环$ k $等效符，并要求$γ$ raded的矩阵环$ r $ yo $ k $超过$ k $的要求很强，跨产品，一个交叉产品，一个倾斜的组环或组环。我们还根据组$γ$在Grothendieck $γ$ -group $ k_0^γ（R）的作用方面表征了这些属性。
If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb Z.$ We formulate properties of the graph $E$ which are equivalent with $L_K(E)$ being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of $L_K(E)$ are also characterized in terms of the pre-ordered group properties of the Grothendieck $\mathbb Z$-group of $L_K(E)$. If $E$ has finitely many vertices, we characterize when $L_K(E)$ is strongly graded in terms of the properties of $K_0^Γ(L_K(E)).$ Our proof also provides an alternative to the known proof of the equivalence $L_K(E)$ is strongly graded if and only if $E$ has no sinks for a finite graph $E.$ We also show that, if unital, the algebra $L_K(E)$ is strongly graded and graded unit-regular if and only if $L_K(E)$ is a crossed product. In the process of showing the main result, we obtain conditions on a group $Γ$ and a $Γ$-graded division ring $K$ equivalent with the requirements that a $Γ$-graded matrix ring $R$ over $K$ is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group $Γ$ on the Grothendieck $Γ$-group $K_0^Γ(R).$
