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== Background ==
The finitegraph 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 <math>E</math> gives a
[[functor]] from this category into the [[natural number]]s <math>\mathbb{N}</math>.
A '''k-graph''' is a natural generalistion of this concept which was introduced in<ref>{{citation |first1=A.|last1=Kumjian |first2=D.A. |last2=Pask |title=[http://nyjm.albany.edu/j/2000/6-1.html ''Higher rank graph C*-algebras''] |journal=[[The New York Journal of Mathematics]] | volume=6 |pages=1–20 | year=2000 }}</ref> by Alex Kumjian and David Pask.
 
== Definition ==
In [[mathematics]], a '''k-graph''' (or higher-rank graph, graph of rank k) is a [[countable]] category <math>\Lambda</math> with domain and codomain maps <math>r</math> and <math>s</math>, together with a functor <math>d : \Lambda \to \mathbb{N}^k</math> which satisfies the following factorisation property: if <math>d ( \lambda ) = m+n</math> then there are unique <math>\mu , \nu \in \Lambda</math> with <math>d ( \mu ) = m , d ( \nu ) = n</math> such that <math>\lambda = \mu \nu</math>.
 
== Examples ==
* It can be shown that a 1-graph is precisely the path category of a directed graph.
* The category <math>T^k</math> consisting of a single object and ''k'' commuting morphisms <math>{f_1,...,f_k}</math>, together with the map <math>d:T^k\to\mathbb{N}^k</math> defined <math>d(f_1^{n_1}...f_k^{n_k})=(n_1 , \ldots , n_k)</math>, is a k-graph.
* Let <math>\Omega_k = \{ (m,n) : m,n \in \mathbb{Z}^k , m \le n \}</math> then <math>\Omega_k</math> is a k-graph when gifted with the structure maps <math>r(m,n)=(m,m)</math>, <math>s(m,n)=(n,n)</math>, <math>(m,n)(n,p)=(m,p)</math> and <math>d(m,n) = n-m</math>.
 
== Notation ==
The notation for k-graphs is borrowed extensively from the corresponding notation for categories:
* For <math>n \in \mathbb{N}^k</math> let <math>\Lambda^n = d^{-1} (n)</math>.
* By the factorisation property it follows that <math>\Lambda^0 = \operatorname{Obj} ( \Lambda )</math>.
* For <math>v,w \in \Lambda^0</math> and <math>X \subseteq \Lambda</math> we have <math>v X = \{ \lambda \in X : r ( \lambda ) = v \}</math>, <math>X w = \{ \lambda \in X : s ( \lambda ) = w \}</math> and <math> v X w = v X \cap X w</math>.
* If <math> 0 < \# v \Lambda^n < \infty</math> for all <math>v \in \Lambda^0</math> and <math>n \in \mathbb{N}^k</math> then <math>\Lambda</math> is said to be row-finite with no sources.
 
== Visualisation - Skeletons ==
A k-graph is best visualised by drawing its 1-skeleton as a [[edge coloring|k-coloured graph]] <math>E=(E^0,E^1,r,s,c)</math> where
<math>E^0 = \Lambda^0</math>, <math>E^1 = \cup_{i=1}^k \Lambda^{e_i}</math>, <math>r,s</math> inherited
from <math>\Lambda</math>
and <math> c: E^1 \to \{ 1 , \ldots , k \}</math> defined by <math>c (e) = i</math>
if and only if <math>e \in \Lambda^{e_i}</math> where <math>e_1 , \ldots , e_n</math> are the canonical
generators for <math>\mathbb{N}^k</math>. The factorisation property in <math>\Lambda</math> for elements
of degree <math>e_i+e_j</math> where <math>i \neq j </math> gives rise to relations between the edges of
<math>E</math>.
 
== C*-algebra ==
As with graph-algebras one may associate a C*-algebra to a k=graph:
 
Let <math>\Lambda</math> be a row-finite k-graph with no sources then a '''Cuntz–Kriger <math>\Lambda</math> family''' in a [[C*-algebra]] B is a collection <math>\{ s_\lambda : \lambda \in \Lambda \}</math> of [[operator (mathematics)|operators]] in B such that
#  <math>s_\lambda s_\mu = s_{\lambda \mu}</math> if <math> \lambda , \mu  , \lambda \mu \in \Lambda</math>;
# <math> \{ s_v : v \in \Lambda^0 \}</math> are mutually orthogonal [[projection (linear algebra)|projections]];
# if <math> d ( \mu ) = d ( \nu )</math> then <math> s_\mu^* s_\nu = \delta_{\mu , \nu} s_{s ( \mu )}</math>;
# <math>s_v = \sum_{\lambda \in v \Lambda^n} s_\lambda s_\lambda^*</math> for all <math>n \in \mathbb{N}^k</math> and <math>v \in \Lambda^0</math>.
 
<math>C^* ( \Lambda )</math> is then the [[universal property|universal]] C*-algebra generated by a Cuntz–Krieger <math>\Lambda</math>-family.
 
== References ==
{{reflist}}
* {{citation|title=Graph algebras |volume=103 |series=CBMS Regional Conference Series in Mathematics |first1=I. |last1=Raeburn |publisher=[[American Mathematical Society]] }}
 
[[Category:C*-algebras|*]]

Latest revision as of 13:46, 17 August 2014

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