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In mathematics, a k-graph (or higher-rank graph, graph of rank k) is a countable category with domain and codomain maps and , together with a functor which satisfies the following factorisation property: if then there are unique with such that .
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.
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 gives a functor from this category into the natural numbers . A k-graph is a natural generalisation of this concept which was introduced in 2000 by Alex Kumjian and David Pask.
- It can be shown that a 1-graph is precisely the path category of a directed graph.
- The category consisting of a single object and k commuting morphisms , together with the map defined , is a k-graph.
- Let then is a k-graph when gifted with the structure maps , , and .
The notation for k-graphs is borrowed extensively from the corresponding notation for categories:
- For let .
- By the factorisation property it follows that .
- For and we have , and .
- If for all and then 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 where , , inherited from and defined by if and only if where are the canonical generators for . The factorisation property in for elements of degree where gives rise to relations between the edges of .
As with graph-algebras one may associate a C*-algebra to a k-graph:
- if ;
- are mutually orthogonal projections;
- if then ;
- for all and .
is then the universal C*-algebra generated by a Cuntz–Krieger -family.
- Raeburn, I., Graph algebras, CBMS Regional Conference Series in Mathematics, 103, American Mathematical Society