CIMPA RESEARCH SCHOOL 2015: A BRIEF INTRODUCTION TO
HIGHER-RANK GRAPHS
ASTRID AN HUEF
Let Λ be a higher-rank graph. The C ∗ -algebra C ∗ (Λ) was defined by Kumjian and Pask
in [6]. The algebraic analogue, the Kumjian-Pask algebra KPR (Λ) over a commutative
ring with identity, was defined in [1]. Why bother? Because they give us a larger class of
examples (see for example, [1, Example 7.1]. Just as with directed graphs, properties of the
higher-rank graph are reflected in their respective algebras – so we can use them to build
examples to order. They are tractable, so we can test theories with them.
Recently, higher-rank graphs have been used to solve an open problem in the classification
program for C ∗ -algebras: Ruiz, Sims and Sørenson used 2-graph algebras in [10] to prove
that every Kirchberg algebra in the UCT class has nuclear dimension 1. They do this by
proving that every Kirchberg algebra in the UCT class is a direct limit of 2-graph algebras.
For simple graph algebras and Leavitt path algebras there is a dichotomy: the algebras
are either AF (ultramatricial) or purely infinite. This is not true for 2-graphs, as illustrated
using “rank-2 Bratteli diagrams” that are cofinal and aperiodic (hence their algebras are
simple by Kumjian and Pask’s simplicity theorem [6, Proposition 4.8]) but are neither AF
(ultramatricial) or purely infinite [8, Theorem 4.3(1)], [1, Theorem 7.10], [2, Proposition 5.7].
Basic definitions and examples. A small1 category C consists of sets C 0 (the objects)
and C ∗ (the morphisms), functions r, s : C ∗ → C 0 , a partially defined product (composition)
(f, g) 7→ f g from {(f, g) ∈ C ∗ × C ∗ : s(f ) = r(g)} to C ∗ , and identity morphisms {ιv ∈ C ∗ :
v ∈ C 0 } such that
(a) r(f g)r(f ) and s(f g) = s(g);
(b) (f g)h = f (gh) when the products are defined;
(c) r(ιv ) = v = s(ιv ), ιv f = f when r(f ) = v and gιv = g when s(g) = v.
We call s(f ), r(f ) the domain and codomain of f , respectively.
A functor F : C → D between small categories C, D is a pair of functions F 0 : C 0 → D0
and F ∗ : C ∗ → D∗ which respect r and s, composition and identity morphism (for example,
F 0 (s(f )) = s(F ∗ (f )) and F ∗ (ιv ) = ιF 0 (v) .
Example 1. Fix k ≥ 1 and consider the semigroup Nk = {n1 , n2 , . . . , nk ) : ni ∈ N}. We
can view Nk a category with 1 object 0 = (0, . . . , 0) and composition mn = m + n =
(n1 + m1 , . . . , mk + nk ) given by addition. The identity morphism at 0 is 0.
Definition 2 (Kumjian-Pask 2000). Let k be a positive integer. A graph of rank k or
k-graph is a countable category Λ = (Λ0 , Λ∗ , r, s) together with a functor d : Λ → Nk , called
the degree map, satisfying the following factorisation property: if λ ∈ Λ and d(λ) = m + n
for some m, n ∈ Nk , then there are unique µ, ν ∈ Λ such that d(µ) = m, d(ν) = n, and
λ = µν.
Date: JUly 7, 2015.
1The “small” means C 0 and C ∗ are sets.
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ASTRID AN HUEF
A k-graph Λ is row-finite if vΛn := r−1 (v) ∩ Λn is finite for every v ∈ Λ0 and n ∈ Nk ; Λ
has no sources if vΛn is nonempty for every v ∈ Λ0 and n ∈ Nk .
Example 3. Let E = (E 0 , E 1 , r, s) be a directed graph. Then the path category P (E) has
object set E 0 , and the morphisms in P (E) from v ∈ E 0 to w ∈ E 0 are finite paths µ with
s(µ) = v and r(µ) = w; composition is defined by concatenation of paths, and the identity
morphisms obtained by viewing the vertices as paths of length 0. With the degree functor
d(µ) = |µ|, the path category (P (E), d) is a 1-graph. Note that P (E)0 is a bijection onto
the paths of length 0 here.
Conventions. Let µ = µ1 . . . µ|µ| and ν be paths in E. To compose µ and ν we need
s(µ) = r(ν). Since we want composition µ ◦ ν to be concatenation µν of paths, we have to
define s(µ) = s(µ|µ| ) and r(µ) = r(µ1 ). Listing a path starting with edges from its range
might seem strange at first, but you get used to it!
Let p, q ∈ Nk . We write p ≤ q if pi ≤ qi for 1 ≤ i ≤ k.
Example 4. Let Ω0k := Nk , Ωk := {(p, q) ∈ Nk × Nk : p ≤ q}, define r, s : Ωk → Ω0k by
r(p, q) := p and s(p, q) := q, define composition by (p, q)(q, r) = (p, r), identity morphism
at p ∈ Ω0k to be (p, p), and d : Ωk → Nk by d(p, q) := q − p. Then Ωk = (Ωk , r, s, d) is a
k-graph. To see what composition does, it helps to draw a picture. Note that p 7→ (p, p) is
a bijection of Ω0k onto the paths of degree 0 here.
Lemma 5. Let Λ be a k-graph. Then ι : Λ0 → Λ∗ is a bijection onto d−1 (0).
Proof. This is a consequence of the factorisation property. Clearly ι is an injection. Let
λ ∈ d−1 (0). Then
ιr(λ) λ = λ = λιs(λ) ,
and the factorisation property with 0 = 0 + 0 implies there exist unique µ, ν ∈ Λ∗ such that
λ = µν and d(µ) = 0 = d(ν). Thus µ = λ and ν = ιs(λ) , and µ = ιr(λ) and ν = λ. In
particular λ = ιr(λ) , and now ι is a bijection onto d−1 (0).
Conventions.
• Motivated by the path category, we call objects vertices and morphisms
paths.
• For n ∈ Nk , write Λn := d−1 (n) for the paths of degree n.
• To simplify notation, we write Λ instead of Λ∗ . We view Λ0 as a subset of Λ using
Lemma 5.
Visualising a k-graph. For simplicity, take k = 2. Let e1 = (1, 0) and e2 = (0, 1) be
the generators of N2 . Think of the objects in Λ0 as the vertices in a directed graph. For
λ ∈ Λe1 , draw a blue (solid) edge from s(λ) to r(λ), and for λ ∈ Λe2 , draw a red (dotted)
edge s(λ) to r(λ). Then E = (Λ0 , Λe1 ∪ Λe2 , r, s) is a coloured graph called the 1-skeleton
of Λ.
For example, the 1-skeleton of the 2-graph Ω2,(3,2) obtained from Ω2 by restricting to
vertices p ≤ (3, 2) is
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HIGHER-RANK GRAPHS
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The path (p, q) with source q and range p is the 2 × 1 rectangle in the top left, and the
different routes ef g, lmg, lih from p to q represent the different factorisations of (p, q) into
blue and red paths.
In other 2-graphs Λ we think of a path λ of degree (2, 1) as a copy of the rectangle
wrapped around the 1-skeleton of Λ in a colour-preserving way. The factorisation property
says that each λ ∈ Λe1 +e2 = Λ(1,1) has both a red-blue and a blue-red factorisation, say
λ = ef = gh where f, g ∈ Λe1 and e, h ∈ Λe2 . We call this a commuting square in the
coloured graph. Note that each red-blue path and each blue-red path occurs in exactly one
commuting square.
This idea also allows us to build 2-graphs:
Proposition 6 (Kumjian-Pask 2000). If E is a 2-coloured graph and C is a collection of
squares in E which contains each red-blue path and each blue-red path exactly once, then
there is a unique 2-graph Λ with Λ0 = E 0 , Λe1 the set of blue edges in E, Λe2 the set of red
edges in E, and Λe1 +e2 = C.
When k ≥ 3, the proposition requires an extra “associativity condition”.
Examples 7. Let Λ be a 2-graph with a single vertex and
(a) one blue edge f and one red edge g. Then C = {gf = f g}.
(b) two blue edges f1 , f2 and one red edge g. Then C = {gf1 = f1 g, gf2 = f2 g} or
C = {gf2 = f1 g, gf1 = f2 g}.
(c) m blue edges f1 , . . . , fm and n red edges {g1 , . . . , gn }. For every permutation θ of
{1, . . . , m} × {1, . . . , n} we get a collection
Cθ = {gi fj = fj0 gi0 : (i0 , j 0 ) = θ(i, j)}.
So we get a 2-graph Λθ where gi fj = fj0 gi0 . Lots of work has been done on the C ∗ algberas C ∗ (Λθ ) of these 1-vertex graphs: Power [9], Davidson-Power-Yang [3, 4],
Davidson-Yang [5], Yang [11]. The following is a sample theorem:
Theorem 8 (Davidson-Yang, 2009). If log m/ log n is irrational, then C ∗ (Λθ ) is
simple and purely infinite.
They have if and only if conditions, but these are complicated. In the proof,
Davidson and Yang characterise when Λθ is periodic, and then use the simplicity
result of Robertson and Sims [7, Theorem 3.1], which says that for a row-finite
k-graph with no sources, C ∗ (Λ) is simple if and only of Λ is aperiodic and cofinal.
The Kumjian-Pask algebra. Please look at [1, §3] for the details of the construction of
the Kumjian-Pask algebra. Let Λ be a k-graph. We let Λ6=0 := {λ ∈ Λ : d(λ) 6= 0}, and for
each λ ∈ Λ6=0 we introduce a ghost path λ∗ ; for v ∈ Λ0 , we define v ∗ := v. We write G(Λ)
for the set of ghost paths, or G(Λ6=0 ) if we wish to exclude vertices. We define d, r and s
on G(Λ) by
d(λ∗ ) = −d(λ), r(λ∗ ) = s(λ), s(λ∗ ) = r(λ);
we then define composition on G(Λ) by setting λ∗ µ∗ = (µλ)∗ for λ, µ ∈ Λ6=0 with r(µ∗ ) =
s(λ∗ ). The factorisation property of Λ induces a similar factorisation property on G(Λ).
Definition 9. Let Λ be a row-finite k-graph without sources and let R be a commutative
ring with 1. A Kumjian-Pask Λ-family (P, S) in an R-algebra A consists of two functions
P : Λ0 → A and S : Λ6=0 ∪ G(Λ6=0 ) → A such that:
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ASTRID AN HUEF
(KP1) {Pv : v ∈ Λ0 } is a family of mutually orthogonal idempotents,
(KP2) for all λ, µ ∈ Λ6=0 with r(µ) = s(λ), we have
Sλ Sµ = Sλµ , Sµ∗ Sλ∗ = S(λµ)∗ , Pr(λ) Sλ = Sλ = Sλ Ps(λ) , Ps(λ) Sλ∗ = Sλ∗ = Sλ∗ Pr(λ) ,
(KP3) for all λ, µ ∈ Λ6=0 with d(λ) = d(µ), we have
Sλ∗ Sµ = δλ,µ Ps(λ) ,
(KP4) for all v ∈ Λ0 and all n ∈ Nk \ {0}, we have
X
Pv =
Sλ Sλ∗ .
λ∈vΛn
It follows from the relations that for each q ≥ d(λ) ∨ d(µ), we have
X
Sλ∗ Sµ =
Sα Sβ ∗ ;
d(λα)=q, λα=µβ
this is what makes the Kumjian-Pask algebras tractable.
Theorem 10 (Aranda Pino-Clark-an Huef-Raeburn, 2013). Let Λ be a row-finite k-graph
without sources, and let R be a commutative ring with 1. Then there is an R-algebra KPR (Λ)
generated by a Kumjian-Pask Λ-family (p, s) such that, whenever (Q, T ) is a Kumjian-Pask
Λ-family in an R-algebra A, there is a unique R-algebra homomorphism πQ,T : KPR (Λ) → A
such that
πQ,T (pv ) = Qv , πQ,T (sλ ) = Tλ , πQ,T (sµ∗ ) = Tµ∗
for v ∈ Λ0 and λ, µ ∈ Λ6=0 . There is a Zk -grading on KPR (Λ) satisfying
(1)
KPR (Λ)n = spanR sλ sµ∗ : λ, µ ∈ Λ and d(λ) − d(µ) = n ,
and we have rpv 6= 0 for v ∈ Λ0 and r ∈ R \ {0}.
Idea of the proof: Consider the free algebra FR (w(X)) on X := Λ0 ∪ Λ6=0 ∪ G(Λ6=0 ). Let
I be the ideal of FR (w(X)) generated by the union of the following sets:
• vw − δv,w v : v, w ∈ Λ0 ;
• λ − µν, λ∗ − ν ∗ µ∗ : λ, µ, ν ∈ Λ6=0 and λ = µν
∪ r(λ)λ − λ, λ − λs(λ), s(λ)λ∗ − λ∗ , λ∗ − λ∗ r(λ) : λ ∈ Λ6=0 ;
• λ∗ µ − δλ,µ s(λ) : λ, µ ∈ Λ6=0 such that d(λ) = d(µ) ;
P
• v − λ∈vΛn λλ∗ : v ∈ Λ0 , n ∈ Nk \ {0} .
Define KPR (Λ) := FR (w(X))/I. Let q : FR (w(X)) → FR (w(X))/I be the quotient map.
Then {pv , sλ , sµ∗ } := {q(v), q(λ), q(µ∗ )} gives a generating Kumjian-Pask Λ-family (p, s) in
KPR (Λ).
The degree functor gives a Zk -grading on F(w(X)) defined by
FR (w(X))n =
n X
w∈w(X)
rw w : rw 6= 0 =⇒ d(w) :=
|w|
X
o
d(wi ) = n
i=1
for n ∈ Zk . The generators of the ideal I are homogeneous, and hence KPR (Λ) is graded by
q(FR (w(X))n ). A bit of work shows this is (1). Finally, we use the free module with basis
the infinite-path space to show there is a Kumjian-Pask family (Q, T ) such that each rQv
is nonzero if r is nonzero.
HIGHER-RANK GRAPHS
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References
[1] G. Aranda Pino, J. Clark, A. an Huef and I. Raeburn, Kumjian-Pask algebras of higher-rank graphs,
Trans. Amer. Math. Soc., 365 (2013), 3613–3641.
[2] L.O. Clark, C. Flynn and A. an Huef, Kumjian-Pask algebras of locally convex higher-rank graphs, J.
Algebra 399 (2014), 445–474.
[3] K.R. Davidson, S.C. Powers and D. Yang, Atomic representations of rank 2 graph algebras, J. Funct.
Anal. 255 (2008), 819–853.
[4] K.R. Davidson, S.C. Powers and D. Yang, Dilation theory for rank 2 graph algebras, J. Operator
Theory.
[5] K.R. Davidson and D. Yang, Periodicity in Rank 2 Graph Algebras, Canad. J. Math. Vol. 61 (2009),
1239–1261.
[6] A. Kumjian and D. Pask, Higher rank graph C ∗ -algebras, New York J. Math. 6 (2000), 1-20.
[7] D.I. Robertson and A. Sims, Simplicity of C ∗ -algebras associated to higher-rank graphs, Bull. London
Math. Soc. 39 (2007), 337–344.
[8] D. Pask, I. Raeburn, M. Rørdam and A. Sims, Rank-two graphs whose C ∗ -algebras are direct limits
of circle algebras, J. Funct. Anal. 239 (2006), 137–178.
[9] S.C. Power, Classifying higher rank analytic Toeplitz algebras, New York J. Math. 13 (2007), 271?298.
[10] E. Ruiz, A.P.W. Sørenson and A. Sims, UCT-Kirchberg algebras have nuclear dimension one, to appear,
arXiv1406.2045[math.OA].
[11] D. Yang, Endomorphisms and modular theory of 2-graph C ∗ -algebras, Indiana Univ. Math. J. 59
(2010), 495–520.