Leavitt path algebras, Cuntz-Krieger algebras, and C^*-graph algebras

Invariant basis number.

A fundamental result in linear algebra is that all bases for a given vector space have the same cardinality. This gives rise to a well-defined notion of dimension. It is not difficult to extend this result to free modules over a commutative ring: all bases for a given free module over a commutative ring have the same cardinality. The same result holds for free modules over noetherian rings. Rings with this property are said to have invariant basis number, IBN for short.

Not all rings have IBN. Given any positive integers $m$ and $n$ there is a ring $R$ such that the free left $R$ -module having a basis of cardinality $m$ is isomorphic to the free left $R$ -module having a basis of cardinality $n$ ; succinctly, $R^m \cong R^n$ .

Rings without invariant basis number.

In 1956 W.G. Leavitt published a short paper entitled Modules without invariant basis number giving examples of this phenomenon. Let $F=k \langle x_1,\ldots,x_n\rangle$ be the free algebra on $n$ variables over a field $k$ . The set of all words in the letters $x_1,\ldots,x_n$ is a $k$ -vector-space basis for $F$ and multiplication is given by concatenation of words. It follows from this that the left ideal ${\mathfrak m}=Fx_1+\cdots+Fx_n$ is a free left $F$ -module with basis $x_1,\ldots,x_n$ . There is an exact sequence $0 \to F^n \to F \to k=F/{\mathfrak m} \to 0$ of left $F$ -modules in which the first map is $(a_1,\ldots,a_n) \mapsto a_1x_1+\cdots+a_nx_n$ . I prefer to write this map as right multiplication by the column vector

$X:=\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}$ .

To get Leavitt’s example, we introduce new variables $y_1,\ldots,y_n$ and form the ring $F \langle y_1,\ldots,y_n \rangle$ modulo the relations:

$X\cdot(y_1,\ldots,y_n)=\begin{pmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0 \\ \vdots && & \vdots \\ 0 & 0 & \ldots & 1\end{pmatrix} \phantom{xx}$ and $\phantom{xx} (y_1,\ldots,y_n)X=1$ .

Call this new ring $R$ . The map $R^n \stackrel{\cdot X}{\longrightarrow} R$ has inverse given by $a \mapsto (ay_1,\ldots,ay_n)$ , which shows that $R^n \cong R$ as left $R$ -modules.

Universal localization.

The construction of $R$ from $F$ is an instance of a general procedure that goes by the name of universal localization. I learned about this from Aidan Schofield’s book Representations of rings over skew fields. The idea is this. Given a ring $S$ and a collection of maps $\Sigma$ between finitely generated projective left $S$ -modules, there is a ring $S_\Sigma$ and a homomorphism $f :S \to S_\Sigma$ with the property that $S_\Sigma \otimes \sigma$ is an isomorphism for all $\sigma \in \Sigma$ and $f :S \to S_\Sigma$ is universal with respect to this property: if $g:S \to T$ is a ring homomorphism such that $T \otimes \sigma$ is an isomorphism for all $\sigma \in \Sigma$ , then $g=hf$ for a unique hmomorphism $h:S_\Sigma \to T$ . The ring $S_\Sigma$ is called the universal localization of $S$ at $\Sigma$ .

In Leavitt’s example, $R$ is the universal localization of $F$ at $\Sigma=\{\cdot X\}$ . The notion of universal localization didn’t emerge until long after Leavitt’s paper but, to me, it is the right context in which to understand Leavitt’s example.

The relations in the localization $R$ .

In preparation for what is to follow I will present $R$ in a slightly different way. First, I will write $x_i^*=y_i$ so $R$ is generated by $x_1,\ldots,x_n$ and $x_1^*,\ldots, x_n^*$ . The relations defining $R$ are $x_ix_i^*=1$ , $x_ix_j^*=0$ if $i \ne j$ , and $x_1^*x_1+\cdots +x_n^*x_n=1$ .

If I had worked with right modules instead of left modules, i.e., if I had written the kernel of the map $F \to F/{\mathfrak m}$ as $x_1F+\cdots+x_nF \to F$ , the resulting universal localization would have been generated by $x_1,\ldots,x_n$ and $x_1^*,\ldots, x_n^*$ subject to the relations $x_i^*x_i=1$ , $x_i^*x_j=0$ if $i \ne j$ , and $x_1x_1^*+\cdots +x_nx_n^*=1$ . Let’s call this ring $R'$ .

The Cuntz algebras ${\mathcal O}_n$ .

In 1977, Joachim Cuntz published Simple C*-Algebras Generated by Isometries. The abstract reads as follows:

We consider the C*-algebra ${\mathcal O}_n$ generated by $n \ge 2$ isometries $S_1,\ldots, S_n$ on an infinite-dimensional Hilbert space, with the property that $S_1S_1^*+\cdots + S_nS_n^*=1$ . It turns out that ${\mathcal O}_n$ has the structure of a crossed product of a finite simple C*-algebra ${\mathcal F}$ by a single endomorphism scaling the trace of ${\mathcal F}$ by $1/n$ . Thus, ${\mathcal O}_n$ is a separable C*-algebra sharing many of the properties of a factor of type $III_\lambda$ with $\lambda=1/n$ . As a consequence we show that ${\mathcal O}_n$ is simple and that its isomorphism type does not depend on the choice of $S_1,\ldots,S_n$ .

An element $p$ in a $C^*$ -algebra is a projection if $p^2=p=p^*$ . An element $S$ in a $C^*$ -algebra is a partial isometry if $S=SS^*S$ . If $S$ is a partial isometry, then $SS^*$ and $S^*S$ are projections. Cuntz is studying the universal $C^*$ -algebra generated by the $S_i$ s so that a full set of relations is given by $S_1S_1^*+\cdots + S_nS_n^*=1$ and the relations $S_i^*S_i=1$ and the range projections $p_i=S_iS_i^*$ are pairwise orthogonal. The orthogonality condition is equivalent to the condition $S_i^*S_j=0$ if $i \ne j$ .

There is an algebra homomorphism $R' \to {\mathcal O}_n$ and the image is a dense subalgebra. For many years I did not understand why Cuntz adopted these relations. Now, through the lens of universal localization, the relations seem natural to me.

The introduction to Cuntz’s paper is easy reading and explains why ${\mathcal O}_n$ is of such interest. The ${\mathcal O}_n$ s settled a number of open problems and exhibited a range of new phenomena. Nevertheless they are relatively easy to work with. Among their important properties is their ubiquity: every simple infinite $C^*$ -algebra contains a subalgebra that has ${\mathcal O}_n$ as a quotient.

The algebras ${\mathcal O}_n$ are themselves simple rings in a very strong way: given a non-zero element $x \in {\mathcal O}_n$ , there are elements $a,b \in {\mathcal O}_n$ such that $axb=1$ . Leavitt had already shown that his algebra $R$ is simple.

Another nice property is that there are, cleverly constructed, inclusions ${\mathcal O}_2 \supset {\mathcal O}_3 \supset \cdots$ . This might not be surprising if you already know there are inclusions $F_2 \supset F_3 \supset \cdots$ where $F_n$ denotes the free algebra on $n$ variables.

The Cuntz algebras are pairwise non-isomorphic because $K_0({\mathcal O}_n) \cong {\mathbb Z}/(n-1)$ .

The path algebra of a quiver

For certain people it is natural when considering some aspect of a free algebra to ask if a similar behavior is exhibited by the path algebra of an arbitrary quiver. The free algebra on $n$ variables is itself a path algebra, the path algebra of the quiver having a single vertex and a $n$ arrows from that vertex to itself. A basis for the path algebra is given by all finite paths in that directed graph.

The free algebra on $n$ variables has an alternative description as the tensor algebra over the field $k$ of an $n$ -dimensional vector space, $k\langle x_1,\ldots,x_n \rangle \cong k \oplus V \oplus V^{\otimes 2} \oplus \cdots$ if $dim(V)=n$ . Path algebras of quivers are also tensor algebras.

Let $Q$ be a finite directed graph, or quiver. Parallel arrows and loops are allowed. The path algebra of $Q$ over a field $k$ , denoted $kQ$ , is the vector space with basis the set of finite paths in $Q$ , including the trivial or lazy paths at each vertex, and multiplication given by concatenation of paths, or zero if the two paths do not concatenate. One now has a difficult decision to make: if $a$ and $b$ are arrows with $a$ ending where $b$ begins should one denote the path “first traverse $a$ then traverse $b$ ” by $ab$ or $ba$ ? The virtue of $ab$ is that one reads from left to right. The virtue of $ba$ is that if we wish to assign vector spaces to vertices and linear maps to arrows the convention for composition of functions is that $ba$ means first do $a$ , then do $b$ . The most popular convention amongst those working on Leavitt path algebras is to write $ab$ . The most popular convention among those working on representation theory of quivers is $ba$ !

Here we adopt the convention that $ab$ means “first traverse $a$ then traverse $b$ ”. If the end of $a$ is not the start of $b$ we define $ab=0$ in $kQ$ . In this way $kQ$ becomes an associative ring.

I will often use the letters $i$ and $j$ denote vertices and will write $e_i$ for the trivial, or lazy, or “do nothing”, path at $i$ . Doing nothing twice is the same as doing nothing once, so $e_ie_i=e_i$ , i.e., $e_i$ is an idempotent. Furthermore, if $a$ is an arrow from $i$ to $j$ , then $e_iae_j=a$ . It follows that the sum of all the $e_i$ s is the identity element in $kQ$ .

As I said above, $kQ$ can be described as a tensor algebra. Let $S$ denote the linear span of the $e_i$ s and write $B$ for the linear span of the arrows. Then $B$ is an $S$ -bimodule and $kQ \cong T_S(B)=S \oplus B \oplus (B \otimes_S B) \oplus \cdots$ , the tensor algebra of $B$ over $S$ .

It is convenient to introduce the notation $s(p)$ for the vertex that is the start of a path $p$ and $t(p)$ for the vertex where $p$ ends.

Leavitt path algebras

To form the Leavitt path algebra $L(Q)$ we first form $\overline{Q}$ , the double of $Q$ : $\overline{Q}$ has the same vertices as $Q$ and for each arrow $a$ we introduce a new arrow $a^*$ , the ghost of $a$ , in the opposite direction, i.e., $a^*$ starts where $a$ ends and ends where $a$ starts; thus, if $a= e_iae_j$ , then $a^*=e_ja^*e_i$ .

We define $L(Q)$ to be the path algebra $k {\overline Q}$ modulo the relations $a^*a=e_{t(a)}$ , $a^*b=0$ if $a \ne b$ for all arrows $a$ and $b$ , and $\sum_{a \in s^{-1}(i)} aa^*=e_i$ for each vertex $i$ at which some arrow ends.

Leavitt path algebras have been intensively studied since they were introduced around 2005.

$C^*$ -graph algebras

Cuntz and Krieger had already introduced the $C^*$ -algebra analogue of Leavitt path algebras in 1980. Let $A$ be an $n \times n$ matrix with entries in $\{0,1\}$ . The Cuntz-Krieger algebra ${\mathcal O}_A$ is the universal $C^*$ -algebra generated by partial isometries $s_1,\ldots,s_n$ subject to the relations $s_i^*s_i=\sum_{j=1}^n A_{ij} s_js_j^*$ and $(s_is_i^*)(s_js_j^*)=0$ for all $i \ne j$ . Of course, $A$ is the incidence matrix of a directed graph $Q$ , and in that case $L(Q)$ is a dense subalgebra of ${\mathcal O}_A$ . If $A_{ij}=1$ for all $i$ and $j$ , then ${\mathcal O}_A \cong {\mathcal O}_n$ .

In 1998, the paper Cuntz-Krieger algebras of directed graphs by Kumjian, Pask, and Raeburn, extended the definition of Cuntz-Krieger algebras to arbitrary directed graphs. If $Q$ is the directed graph, they write $C^*(Q)$ for the universal $C^*$ -algebra generated by mutually orthogonal projections $p_i$ , one for each vertex $i$, and partial isometries $a$ , one for each arrow, subject to the relations

$a^*a = p_{t(a)} \phantom{xx}$ and $p_i=\sum_{a \in s^{-1}(i)} aa^*$ .

Raeburn’s 2005 CBMS lectures Graph Algebras gives a comprehensive account of the subject up to 2005. The document Graph algebras: bridging the gap between analysis and algebra is a useful introduction to graph algebras and Leavitt path algebras that is aimed at algebraists.

I have restricted my discussion to finite directed graphs but almost everything I have said extends in a reasonable way to arbitrary directed graphs.

Why am I talking about this?

I’m in Banff attending the meeting Linking representation theory, singularity theory and non-commutative algebraic geometry. I’m eager to hear Xiao-Wu Chen’s talk today on his paper Irreducible representations of Leavitt path algebras. I have been reading it this week. My interest arises from my paper Category equivalences involving graded modules over path algebras of quivers, where I show that a certain quotient category, $QGr(kQ)$ , of the category of graded modules over the path algebra of a quiver (directed graph) $Q$ having a finite number of vertices and arrows is equivalent to the category of graded modules over the Leavitt path algebra $L(Q^\circ)$ associated to the quiver $Q^\circ$ that is obtained by successively removing all sinks and sources from $Q$ .

Leavitt path algebras, Cuntz-Krieger algebras, and C^*-graph algebras

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