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 $d : \Lambda \to \mathbb{N}^k$ which satisfies the following factorisation property: if $d ( \lambda ) = m+n$ then there are unique $\mu , \nu \in \Lambda$ with $d ( \mu ) = m , d ( \nu ) = n$ such that $\lambda = \mu \nu$.

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 $T^k$ consisting of a single object and k commuting morphisms ${f_1,...,f_k}$, together with the map $d:T^k\to\mathbb{N}^k$ defined $d(f_1^{n_1}...f_k^{n_k})=(n_1 , \ldots , n_k)$, is a k-graph.
• Let $\Omega_k = \{ (m,n) : m,n \in \mathbb{Z}^k , m \le n \}$ then $\Omega_k$ is a k-graph when gifted with the structure maps $r(m,n)=(m,m)$, $s(m,n)=(n,n)$, $(m,n)(n,p)=(m,p)$ and $d(m,n) = n-m$.

Notation

The notation for k-graphs is borrowed extensively from the corresponding notation for categories:

• For $n \in \mathbb{N}^k$ let $\Lambda^n = d^{-1} (n)$.
• By the factorisation property it follows that $\Lambda^0 = \operatorname{Obj} ( \Lambda )$.
• For $v,w \in \Lambda^0$ and $X \subseteq \Lambda$ we have $v X = \{ \lambda \in X : r ( \lambda ) = v \}$, $X w = \{ \lambda \in X : s ( \lambda ) = w \}$ and $v X w = v X \cap X w$.
• If $0 < \# v \Lambda^n < \infty$ for all $v \in \Lambda^0$ and $n \in \mathbb{N}^k$ 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 $E=(E^0,E^1,r,s,c)$ where $E^0 = \Lambda^0$, $E^1 = \cup_{i=1}^k \Lambda^{e_i}$, $r,s$ inherited from $\Lambda$ and $c: E^1 \to \{ 1 , \ldots , k \}$ defined by $c (e) = i$ if and only if $e \in \Lambda^{e_i}$ where $e_1 , \ldots , e_n$ are the canonical generators for $\mathbb{N}^k$. The factorisation property in $\Lambda$ for elements of degree $e_i+e_j$ where $i \neq j$ 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 $\{ s_\lambda : \lambda \in \Lambda \}$ of operators in B such that

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

$C^* ( \Lambda )$ 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