Free semigroupoid algebras and the first cohomology groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915417643024384 |
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| author | Huang, Linzhe Ma, Minghui |
| author_facet | Huang, Linzhe Ma, Minghui |
| contents | 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 $δ$ from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is of the form $δ=δ_T$ for some $T\in\mathfrak{L}_G$ with $\|T\|\leqslant\|δ\|$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free semigroupoid algebras and the first cohomology groups Huang, Linzhe Ma, Minghui Operator Algebras Functional Analysis 46L10, 46K50, 47B47 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 $δ$ from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is of the form $δ=δ_T$ for some $T\in\mathfrak{L}_G$ with $\|T\|\leqslant\|δ\|$. 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. |
| title | Free semigroupoid algebras and the first cohomology groups |
| topic | Operator Algebras Functional Analysis 46L10, 46K50, 47B47 |
| url | https://arxiv.org/abs/2507.22508 |