k-graph C*-algebra

In mathematics, a k-graph (or higher-rank graph, graph of rank k) is a countable category $\Lambda$ with domain and codomain maps $r$ and $s$, together with a functor ${\displaystyle d:\Lambda \to \mathbb {N} ^{k}}$ which satisfies the following factorisation property: if ${\displaystyle d(\lambda )=m+n}$ then there are unique ${\displaystyle \mu ,\nu \in \Lambda }$ with ${\displaystyle d(\mu )=m,d(\nu )=n}$ such that ${\displaystyle \lambda =\mu \nu }$.

Aside from its category theory definition, one can think of k-graphs as higher dimensional analogue of directed graphs (digraphs). k- here signifies the number of "colors" of edges that are involved in the graph. If k=1, k-graph is just a regular directed graph. If k=2, there are two different colors of edges involved in the graph and additional factorization rules of 2-color equivalent classes should be defined. The factorization rule on k-graph skeleton is what distinguishes one k-graph defined on the same skeleton from another k-graph. k- can be any natural number greater than or equal to 1.

The reason k-graphs were first introduced by Kumjian, Pask et. al. was to create examples of C*-algebra from them. k-graphs consist of two parts: skeleton and factorization rules defined on the given skeleton. Once k-graph is well-defined, one can define functions called 2-cocycles on each graph, and C*-algebras can be built from k-graphs and 2-cocycles. k-graphs are relatively simple to understand from graph theory perspective, yet just complicated enough to reveal different interesting properties in the C*-algebra level. The properties such as homotopy and cohomology on the 2-cocycles defined on k-graphs have implications to C*-algebra and K-theory research efforts. No other known use of k-graphs exist to this day. k-graphs are studied solely for the purpose of creating C*-algebras from them.

Background

The finite graph theory in a directed graph form a mathematics category under concatenation called the free object category (which is generated by a graph). The length of a path in $E$ gives a functor from this category into the natural numbers $\mathbb {N}$. A k-graph is a natural generalisation of this concept which was introduced in 2000 by Alex Kumjian and David Pask.^[1]

Examples

• It can be shown that a 1-graph is precisely the path category of a directed graph.
• The category ${\displaystyle T^{k}}$ consisting of a single object and k commuting morphisms ${\displaystyle {f_{1},...,f_{k}}}$, together with the map ${\displaystyle d:T^{k}\to \mathbb {N} ^{k}}$ defined ${\displaystyle d(f_{1}^{n_{1}}...f_{k}^{n_{k}})=(n_{1},\ldots ,n_{k})}$, is a k-graph.
• Let ${\displaystyle \Omega _{k}=\{(m,n):m,n\in \mathbb {Z} ^{k},m\leq n\}}$ then $\Omega_k$ is a k-graph when gifted with the structure maps ${\displaystyle r(m,n)=(m,m)}$, ${\displaystyle s(m,n)=(n,n)}$, ${\displaystyle (m,n)(n,p)=(m,p)}$ and ${\displaystyle d(m,n)=n-m}$.

Notation

The notation for 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 ${\displaystyle \Lambda ^{0}=\operatorname {Obj} (\Lambda )}$.
• 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 $\Lambda$ is said to be row-finite with no sources.

Visualisation - Skeletons

A k-graph is best visualised by drawing its 1-skeleton as a k-coloured graph ${\displaystyle E=(E^{0},E^{1},r,s,c)}$ where ${\displaystyle E^{0}=\Lambda ^{0}}$, ${\displaystyle E^{1}=\cup _{i=1}^{k}\Lambda ^{e_{i}}}$, $r,s$ inherited from $\Lambda$ and ${\displaystyle c:E^{1}\to \{1,\ldots ,k\}}$ defined by ${\displaystyle c(e)=i}$ if and only if ${\displaystyle e\in \Lambda ^{e_{i}}}$ where ${\displaystyle e_{1},\ldots ,e_{n}}$ are the canonical generators for ${\mathbb {N}}^{k}$. The factorisation property in $\Lambda$ for elements of degree ${\displaystyle e_{i}+e_{j}}$ where ${\displaystyle i\neq j}$ gives rise to relations between the edges of $E$.

C*-algebra

As with graph-algebras one may associate a C*-algebra to a k-graph:

Let $\Lambda$ be a row-finite k-graph with no sources then a Cuntz–Krieger $\Lambda$ family in a C*-algebra B is a collection ${\displaystyle \{s_{\lambda }:\lambda \in \Lambda \}}$ of operators in B such that

1. ${\displaystyle s_{\lambda }s_{\mu }=s_{\lambda \mu }}$ if ${\displaystyle \lambda ,\mu ,\lambda \mu \in \Lambda }$;
2. ${\displaystyle \{s_{v}:v\in \Lambda ^{0}\}}$ are mutually orthogonal projections;
3. if ${\displaystyle d(\mu )=d(\nu )}$ then ${\displaystyle s_{\mu }^{*}s_{\nu }=\delta _{\mu ,\nu }s_{s(\mu )}}$;
4. ${\displaystyle s_{v}=\sum _{\lambda \in v\Lambda ^{n}}s_{\lambda }s_{\lambda }^{*}}$ for all ${\displaystyle n\in \mathbb {N} ^{k}}$ and ${\displaystyle v\in \Lambda ^{0}}$.

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

References

1. Kumjian, A.; Pask, D.A. (2000), "Higher rank graph C*-algebras", The New York Journal of Mathematics, 6: 1–20

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