[Paper Review] The center of a Kumjian-Pask algebra

Jonathan H. Brown, Astrid an Huef|arXiv (Cornell University)|Sep 12, 2012

Advanced Operator Algebra Research3 references3 citations

TL;DR

This paper characterizes the center of Kumjian-Pask algebras over a commutative ring with 1, showing that for a basically simple algebra, the center is either zero or isomorphic to the base ring. It further proves that a Kumjian-Pask algebra is commutative if and only if the underlying k-graph is a disjoint union of copies of $\mathbb{N}^k$, in which case the algebra is isomorphic to a direct sum of Laurent polynomial rings in $k$ variables.

ABSTRACT

The Kumjian-Pask algebras are path algebras associated to higher-rank graphs, and generalize the Leavitt path algebras. We study the center of simple Kumjian-Pask algebras and characterize commutative Kumjian-Pask algebras.

Motivation & Objective

• To characterize the center of simple and basically simple Kumjian-Pask algebras over a commutative ring with 1.
• To determine when a Kumjian-Pask algebra is commutative, generalizing known results for Leavitt path algebras.
• To establish a structural classification of commutative Kumjian-Pask algebras via graph-theoretic properties of the underlying k-graph.
• To use the embedding into the associated C*-algebra and the Dauns-Hofmann theorem to analyze the center, particularly over $\mathbb{C}$.

Proposed method

• Utilizes the embedding of the Kumjian-Pask algebra $\mathrm{KP}_R(\Lambda)$ into its associated $C^*$-algebra $C^*(\Lambda)$ to analyze the center via topological methods.
• Applies the Dauns-Hofmann theorem to relate the center of $\mathrm{KP}_\mathbb{C}(\Lambda)$ to the structure of the spectrum of $C^*(\Lambda)$, showing it is either $\{0\}$ or $\mathbb{C}$.
• Employs the graded uniqueness theorem and universal property of Kumjian-Pask algebras to establish isomorphisms and injectivity of induced maps.
• Analyzes the graph-theoretic conditions of cofinality and aperiodicity to deduce properties of elements in the center.
• Uses the universal property and relations in the Kumjian-Pask algebra to derive constraints on commutativity and injectivity of the range map $r$.
• Applies the decomposition of a k-graph as a disjoint union of subgraphs to reduce the problem to the case of $\mathbb{N}^k$, where the algebra is known to be a Laurent polynomial ring.

Experimental results

Research questions

• RQ1 When is the center of a Kumjian-Pask algebra trivial or isomorphic to the base ring $R$?
• RQ2 What graph-theoretic conditions on a k-graph $\Lambda$ ensure that $\mathrm{KP}_R(\Lambda)$ is commutative?
• RQ3 How does the structure of the center of $\mathrm{KP}_R(\Lambda)$ relate to the topological properties of the associated $C^*$-algebra $C^*(\Lambda)$?
• RQ4 Under what conditions is $\mathrm{KP}_R(\Lambda)$ isomorphic to a direct sum of Laurent polynomial rings in $k$ variables?
• RQ5 What characterizes the class of k-graphs $\Lambda$ for which $\mathrm{KP}_R(\Lambda)$ is commutative?

Key findings

• The center of a simple Kumjian-Pask algebra over $\mathbb{C}$ is either $\{0\}$ or isomorphic to $\mathbb{C}$, as deduced from the Dauns-Hofmann theorem and the embedding into $C^*(\Lambda)$.
• For a basically simple Kumjian-Pask algebra $\mathrm{KP}_R(\Lambda)$, the center is either zero or isomorphic to the base ring $R$, generalizing a result for Leavitt path algebras.
• A Kumjian-Pask algebra $\mathrm{KP}_R(\Lambda)$ is commutative if and only if the range map $r$ is injective on each $\Lambda^n$ and $r=s$ on $\Lambda$, which implies $\Lambda$ is a disjoint union of copies of $\mathbb{N}^k$.
• If $\Lambda$ is a disjoint union of copies of $\mathbb{N}^k$, then $\mathrm{KP}_R(\Lambda) \cong \bigoplus_{v \in \Lambda^0} R[x_1, x_1^{-1}, \dots, x_k, x_k^{-1}]$, a direct sum of Laurent polynomial rings.
• The algebra $\mathrm{KP}_R(\mathbb{N}^k)$ is isomorphic to the Laurent polynomial ring $R[x_1, x_1^{-1}, \dots, x_k, x_k^{-1}]$, and this structure underlies the classification of commutative Kumjian-Pask algebras.
• The proof shows that the graph properties of cofinality and aperiodicity are essential in determining the center, with cofinality ensuring the center is nontrivial and aperiodicity ruling out nontrivial central elements.