标题： 自由半群范畴代数和第一上同调群 显示英文标题

标题： Free semigroupoid algebras and the first cohomology groups

摘要： 本文研究了可数或不可数有向图 $G$ 的自由半群oid代数 $\mathfrak{L}_G$及其范数闭合版本，即张量代数 $\mathcal{A}_G$。 我们首先证明了当 $G$ 是强连通时， $\mathfrak{L}_G$的弱Dixmier逼近定理。 使用该定理，我们证明如果$G$的每个连通分支都是强连通的，那么从$\mathcal{A}_G$到$\mathfrak{L}_G$的每个有界导子$\delta$都是$\delta=\delta_T$的形式，其中$T\in\mathfrak{L}_G$满足$\|T\|\leqslant\|\delta\|$。 对于任何有限有向图$G$，我们还证明，第一上同调群$H^1(\mathcal{A}_G,\mathfrak{L}_G)$为零当且仅当$G$的每个连通分支要么是强连通的，要么是果实树。 为了处理无限有向图，我们引入了交替数并提出了\Cref{conj intro-in-tree}。 假设$G$的每个连通分支都不是强连通的。 我们证明，如果每个从$\mathcal{A}_G$到$\mathfrak{L}_G$的有界导子都是内导子，那么$G$的每个连通分支都是广义水果树，且$G$的交错数$A(G)$是有限的。 如果该猜想成立，那么逆命题也成立。 最后，我们提供了一些自由半群范畴代数及其非平凡的一阶上同调群的例子。 显示英文摘要

摘要： This paper investigates derivations of the free semigroupoid algebra $\mathfrak{L}_G$ of a countable or uncountable directed graph $G$ and its norm-closed version, the tensor algebra $\mathcal{A}_G$. We first prove a weak Dixmier approximation theorem for $\mathfrak{L}_G$ when $G$ is strongly connected. Using the theorem, we show that if every connected component of $G$ is strongly connected, then every bounded derivation $\delta$ from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is of the form $\delta=\delta_T$ for some $T\in\mathfrak{L}_G$ with $\|T\|\leqslant\|\delta\|$. For any finite directed graph $G$, we also show that the first cohomology group $H^1(\mathcal{A}_G,\mathfrak{L}_G)$ vanishes if and only if every connected component of $G$ is either strongly connected or a fruit tree. To handle infinite directed graphs, we introduce the alternating number and propose \Cref{conj intro-in-tree}. Suppose every connected component of $G$ is not strongly connected. We show that if every bounded derivation from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is inner, then every connected component of $G$ is a generalized fruit tree and the alternating number $A(G)$ of $G$ is finite. The converse is also true if the conjecture holds. Finally, we provide some examples of free semigroupoid algebras together with their nontrivial first cohomology groups.