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In mathematics, for ${\displaystyle k\in \mathbb {N} }$, a ${\displaystyle k}$-graph (also known as a higher-rank graph or graph of rank ${\displaystyle k}$) is a countable category ${\displaystyle \Lambda }$ together with a functor ${\displaystyle d:\Lambda \to \mathbb {N} ^{k}}$, called the degree map, which satisfy the following factorization property:

if ${\displaystyle \lambda \in \Lambda }$ and ${\displaystyle m,n\in \mathbb {N} ^{k}}$ are such that ${\displaystyle d(\lambda )=m+n}$, then there exist unique ${\displaystyle \mu ,\nu \in \Lambda }$ such that ${\displaystyle d(\mu )=m}$, ${\displaystyle d(\nu )=n}$, and ${\displaystyle \lambda =\mu \nu }$.

An immediate consequence of the factorization property is that morphisms in a ${\displaystyle k}$-graph can be factored in multiple ways: there are also unique ${\displaystyle \mu ',\nu '\in \Lambda }$ such that ${\displaystyle d(\mu ')=m}$, ${\displaystyle d(\nu ')=n}$, and ${\displaystyle \mu \nu =\lambda =\nu '\mu '}$.

A 1-graph is just the path category of a directed graph. In this case the degree map takes a path to its length. By extension, ${\displaystyle k}$-graphs can be considered higher-dimensional analogs of directed graphs.

Another way to think about a ${\displaystyle k}$-graph is as a ${\displaystyle k}$-colored directed graph together with additional information to record the factorization property. The ${\displaystyle k}$-colored graph underlying a ${\displaystyle k}$-graph is referred to as its skeleton. Two ${\displaystyle k}$-graphs can have the same skeleton but different factorization rules.

Kumjian and Pask originally introduced ${\displaystyle k}$-graphs as a generalization of a construction of Robertson and Steger. By considering representations of ${\displaystyle k}$-graphs as bounded operators on Hilbert space, they have since become a tool for constructing interesting C*-algebras whose structure reflects the factorization rules. Some compact quantum groups like ${\displaystyle SU_{q}(3)}$ can be realised as the ${\displaystyle C^{*}}$-algebras of ${\displaystyle k}$-graphs. There is also a close relationship between ${\displaystyle k}$-graphs and strict factorization systems in category theory.

Notation

The notation for ${\displaystyle k}$-graphs is borrowed extensively from the corresponding notation for categories:

• For ${\displaystyle n\in \mathbb {N} ^{k}}$ let ${\displaystyle \Lambda ^{n}=d^{-1}(n)}$. By the factorisation property it follows that ${\displaystyle \Lambda ^{0}=\operatorname {Obj} (\Lambda )}$.

• There are maps ${\displaystyle s:\Lambda \to \Lambda ^{0}}$ and ${\displaystyle r:\Lambda \to \Lambda ^{0}}$ which take a morphism ${\displaystyle \lambda \in \Lambda }$ to its source ${\displaystyle s(\lambda )}$ and its range ${\displaystyle r(\lambda )}$.

• For ${\displaystyle v,w\in \Lambda ^{0}}$ and ${\displaystyle X\subseteq \Lambda }$ we have ${\displaystyle vX=\{\lambda \in X:r(\lambda )=v\}}$, ${\displaystyle Xw=\{\lambda \in X:s(\lambda )=w\}}$ and ${\displaystyle vXw=vX\cap Xw}$.

• If ${\displaystyle 0<\#v\Lambda ^{n}<\infty }$ for all ${\displaystyle v\in \Lambda ^{0}}$ and ${\displaystyle n\in \mathbb {N} ^{k}}$ then ${\displaystyle \Lambda }$ is said to be row-finite with no sources.

Skeletons

A ${\displaystyle k}$-graph ${\displaystyle \Lambda }$ can be visualized via its skeleton. Let ${\displaystyle e_{1},\ldots ,e_{n}}$ be the canonical generators for ${\displaystyle \mathbb {N} ^{k}}$. The idea is to think of morphisms in ${\displaystyle \Lambda ^{e_{i}}=d^{-1}(e_{i})}$ as being edges in a directed graph of a color indexed by ${\displaystyle i}$.

To be more precise, the skeleton of a ${\displaystyle k}$-graph ${\displaystyle \Lambda }$ is a k-colored directed graph ${\displaystyle E=(E^{0},E^{1},r,s,c)}$ with vertices ${\displaystyle E^{0}=\Lambda ^{0}}$, edges ${\displaystyle E^{1}=\cup _{i=1}^{k}\Lambda ^{e_{i}}}$, range and source maps inherited from ${\displaystyle \Lambda }$, and a color map ${\displaystyle c:E^{1}\to \{1,\ldots ,k\}}$ defined by ${\displaystyle c(e)=i}$ if and only if ${\displaystyle e\in \Lambda ^{e_{i}}}$.

The skeleton of a ${\displaystyle k}$-graph alone is not enough to recover the ${\displaystyle k}$-graph. The extra information about factorization can be encoded in a complete and associative collection of commuting squares. In particular, for each ${\displaystyle i\neq j}$ and ${\displaystyle e,f\in E^{1}}$ with ${\displaystyle c(e)=i}$ and ${\displaystyle c(f)=j}$, there must exist unique ${\displaystyle e',f'\in E^{1}}$ with ${\displaystyle c(e')=i}$, ${\displaystyle c(f')=j}$, and ${\displaystyle ef=f'e'}$ in ${\displaystyle \Lambda }$. A different choice of commuting squares can yield a distinct ${\displaystyle k}$-graph with the same skeleton.

Examples

• A 1-graph is precisely the path category of a directed graph. If ${\displaystyle \lambda }$ is a path in the directed graph, then ${\displaystyle d(\lambda )}$ is its length. The factorization condition is trivial: if ${\displaystyle \lambda }$ is a path of length ${\displaystyle m+n}$ then let ${\displaystyle \mu }$ be the initial subpath of length ${\displaystyle m}$ and let ${\displaystyle \nu }$ be the final subpath of length ${\displaystyle n}$.

• The monoid ${\displaystyle \mathbb {N} ^{k}}$ can be considered as a category with one object. The identity on ${\displaystyle \mathbb {N} ^{k}}$ give a degree map making ${\displaystyle \mathbb {N} ^{k}}$ into a ${\displaystyle k}$-graph.

• Let ${\displaystyle \Omega _{k}=\{(m,n):m,n\in \mathbb {Z} ^{k},m\leq n\}}$. Then ${\displaystyle \Omega _{k}}$ is a category with range map ${\displaystyle r(m,n)=(m,m)}$, source map ${\displaystyle s(m,n)=(n,n)}$, and composition ${\displaystyle (m,n)(n,p)=(m,p)}$. Setting ${\displaystyle d(m,n)=n-m}$ gives a degree map. The factorization rule is given as follows: if ${\displaystyle d(m,n)=p+q}$ for some ${\displaystyle p,q\in \mathbb {N} ^{k}}$, then ${\displaystyle (m,n)=(m,m+q)(m+q,n)}$ is the unique factorization.

C*-algebras of k-graphs

Just as a graph C*-algebra can be associated to a directed graph, a universal C*-algebra can be associated to a ${\displaystyle k}$-graph.

Let ${\displaystyle \Lambda }$ be a row-finite ${\displaystyle k}$-graph with no sources then a Cuntz–Krieger ${\displaystyle \Lambda }$-family or a represenentaion of ${\displaystyle \Lambda }$ in a C*-algebra B is a map ${\displaystyle S\colon \Lambda \to B}$ such that

1. ${\displaystyle \{S_{v}:v\in \Lambda ^{0}\}}$ is a collection of mutually orthogonal projections;
2. ${\displaystyle S_{\lambda }S_{\mu }=S_{\lambda \mu }}$ for all ${\displaystyle \lambda ,\mu \in \Lambda }$ with ${\displaystyle s(\lambda )=r(\mu )}$;
3. ${\displaystyle S_{\mu }^{*}S_{\mu }=S_{s(\mu )}}$ for all ${\displaystyle \mu \in \Lambda }$; and
4. ${\displaystyle S_{v}=\sum _{\lambda \in v\Lambda ^{n}}S_{\lambda }S_{\lambda }^{*}}$ for all ${\displaystyle n\in \mathbb {N} ^{k}}$ and ${\displaystyle v\in \Lambda ^{0}}$.

The algebra ${\displaystyle C^{*}(\Lambda )}$ is the universal C*-algebra generated by a Cuntz–Krieger ${\displaystyle \Lambda }$-family.

See also

• Graph C*-algebra

References

1. Kumjian, A.; Pask, D.A. (2000), "Higher rank graph C*-algebras", The New York Journal of Mathematics, 6: 1–20
2. Giselsson, O. (2023), "Quantum SU(3) as the C*-algebra of a 2-Graph", arXiv math.OA
3. Sims, A., Lecture notes on higher-rank graphs and their C*-algebras (PDF)

• Raeburn, I., Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, American Mathematical Society

• C*-algebras