Free semigroupoid algebras and the first cohomology groups

TL;DR

This paper studies derivations and cohomology of free semigroupoid algebras associated with directed graphs, establishing conditions for derivation innerness and cohomology vanishing, and introducing the alternating number for infinite graphs.

Contribution

It introduces new results on derivations and cohomology of free semigroupoid algebras, including a weak Dixmier approximation theorem and a conjecture involving the alternating number for infinite graphs.

Findings

Bounded derivations are inner under certain graph conditions.

First cohomology vanishes if each component is strongly connected or a fruit tree.

Examples demonstrate nontrivial first cohomology groups.

Abstract

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.