Leavitt path algebra

In mathematics, a Leavitt path algebra is a universal algebra constructed from a directed graph. Leavitt path algebras generalize Leavitt algebras and may be considered as algebraic analogues of graph C*-algebras.

Leavitt path algebras were simultaneously introduced in 2005 by Gene Abrams and Gonzalo Aranda Pino^[1] as well as by Pere Ara, María Moreno, and Enrique Pardo,^[2] with neither of the two groups aware of the other's work.^[3] Leavitt path algebras have been investigated by dozens of mathematicians since their introduction, and in 2020 Leavitt path algebras were added to the Mathematics Subject Classification with code 16S88 under the general discipline of Associative Rings and Algebras.^[4]

The basic reference is the book Leavitt Path Algebras.^[5]

The theory of Leavitt path algebras uses terminology for graphs similar to that of C*-algebraists, which differs slightly from that used by graph theorists. The term graph is typically taken to mean a directed graph ${\displaystyle E=(E^{0},E^{1},r,s)}$ consisting of a countable set of vertices ${\displaystyle E^{0}}$, a countable set of edges ${\displaystyle E^{1}}$, and maps ${\displaystyle r,s:E^{1}\rightarrow E^{0}}$ identifying the range and source of each edge, respectively. A vertex ${\displaystyle v\in E^{0}}$ is called a sink when ${\displaystyle s^{-1}(v)=\emptyset }$; i.e., there are no edges in ${\displaystyle E}$ with source ${\displaystyle v}$. A vertex ${\displaystyle v\in E^{0}}$ is called an infinite emitter when ${\displaystyle s^{-1}(v)}$ is infinite; i.e., there are infinitely many edges in ${\displaystyle E}$ with source ${\displaystyle v}$. A vertex is called a singular vertex if it is either a sink or an infinite emitter, and a vertex is called a regular vertex if it is not a singular vertex. Note that a vertex ${\displaystyle v}$ is regular if and only if the number of edges in ${\displaystyle E}$ with source ${\displaystyle v}$ is finite and nonzero. A graph is called row-finite if it has no infinite emitters; i.e., if every vertex is either a regular vertex or a sink.

A path is a finite sequence of edges ${\displaystyle e_{1}e_{2}\ldots e_{n}}$ with ${\displaystyle r(e_{i})=s(e_{i+1})}$ for all ${\displaystyle 1\leq i\leq n-1}$. An infinite path is a countably infinite sequence of edges ${\displaystyle e_{1}e_{2}\ldots }$ with ${\displaystyle r(e_{i})=s(e_{i+1})}$ for all ${\displaystyle i\in \mathbb {N} }$. A cycle is a path ${\displaystyle e_{1}e_{2}\ldots e_{n}}$ with ${\displaystyle r(e_{n})=s(e_{1})}$, and an exit for a cycle ${\displaystyle e_{1}e_{2}\ldots e_{n}}$ is an edge ${\displaystyle f\in E^{1}}$ such that ${\displaystyle s(f)=s(e_{i})}$ and ${\displaystyle f\neq e_{i}}$ for some ${\displaystyle 1\leq i\leq n}$. A cycle ${\displaystyle e_{1}e_{2}\ldots e_{n}}$ is called a simple cycle if ${\displaystyle s(e_{i})\neq s(e_{1})}$ for all ${\displaystyle 2\leq i\leq n}$.

The following are two important graph conditions that arise in the study of Leavitt path algebras.

Condition (L): Every cycle in the graph has an exit.

Condition (K): There is no vertex in the graph that is on exactly one simple cycle. Equivalently, a graph satisfies Condition (K) if and only if each vertex in the graph is either on no cycles or on two or more simple cycles.

Fix a field ${\displaystyle K}$. A Cuntz–Krieger ${\displaystyle E}$-family is a collection ${\displaystyle \{s_{e}^{*},s_{e},p_{v}:e\in E^{1},v\in E^{0}\}}$ in a ${\displaystyle K}$-algebra such that the following three relations (called the Cuntz–Krieger relations) are satisfied:

(CK0) ${\displaystyle p_{v}p_{w}={\begin{cases}p_{v}&{\text{if }}v=w\\0&{\text{if }}v\neq w\end{cases}}\quad }$ for all ${\displaystyle v,w\in E^{0}}$,

(CK1) ${\displaystyle s_{e}^{*}s_{f}={\begin{cases}p_{r(e)}&{\text{if }}e=f\\0&{\text{if }}e\neq f\end{cases}}\quad }$ for all ${\displaystyle e,f\in E^{1}}$,

(CK2) ${\displaystyle p_{v}=\sum _{s(e)=v}s_{e}s_{e}^{*}}$ whenever ${\displaystyle v}$ is a regular vertex, and

(CK3) ${\displaystyle p_{s(e)}s_{e}=s_{e}}$ for all ${\displaystyle e\in E^{1}}$.

The Leavitt path algebra corresponding to ${\displaystyle E}$, denoted by ${\displaystyle L_{K}(E)}$, is defined to be the ${\displaystyle K}$-algebra generated by a Cuntz–Krieger ${\displaystyle E}$-family that is universal in the sense that whenever ${\displaystyle \{t_{e},t_{e}^{*},q_{v}:e\in E^{1},v\in E^{0}\}}$ is a Cuntz–Krieger ${\displaystyle E}$-family in a ${\displaystyle K}$-algebra ${\displaystyle A}$ there exists a ${\displaystyle K}$-algebra homomorphism ${\displaystyle \phi :L_{K}(E)\to A}$ with ${\displaystyle \phi (s_{e})=t_{e}}$ for all ${\displaystyle e\in E^{1}}$, ${\displaystyle \phi (s_{e}^{*})=t_{e}^{*}}$ for all ${\displaystyle e\in E^{1}}$, and ${\displaystyle \phi (p_{v})=q_{v}}$ for all ${\displaystyle v\in E^{0}}$.

We define ${\displaystyle p_{v}^{*}:=p_{v}}$ for ${\displaystyle v\in E^{0}}$, and for a path ${\displaystyle \alpha :=e_{1}\ldots e_{n}}$ we define ${\displaystyle s_{\alpha }:=s_{e_{1}}\ldots s_{e_{n}}}$ and ${\displaystyle s_{\alpha }^{*}:=s_{e_{n}}^{*}\ldots s_{e_{1}}^{*}}$. Using the Cuntz–Krieger relations, one can show that

${\displaystyle L_{K}(E)=\operatorname {span} _{K}\{s_{\alpha }s_{\beta }^{*}:\alpha {\text{ and }}\beta {\text{ are paths in }}E\}.}$

Thus a typical element of ${\displaystyle L_{K}(E)}$ has the form ${\displaystyle \sum _{i=1}^{n}\lambda _{i}s_{\alpha _{i}}s_{\beta _{i}}^{*}}$ for scalars ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}\in K}$ and paths ${\displaystyle \alpha _{1},\ldots ,\alpha _{n},\beta _{1},\ldots ,\beta _{n}}$ in ${\displaystyle E}$. If ${\displaystyle K}$ is a field with an involution ${\displaystyle \lambda \mapsto {\overline {\lambda }}}$ (e.g., when ${\displaystyle K=\mathbb {C} }$), then one can define a *-operation on ${\displaystyle L_{K}(E)}$ by ${\displaystyle \sum _{i=1}^{n}\lambda _{i}s_{\alpha _{i}}s_{\beta _{i}}^{*}\mapsto \sum _{i=1}^{n}{\overline {\lambda _{i}}}s_{\beta _{i}}s_{\alpha _{i}}^{*}}$ that makes ${\displaystyle L_{K}(E)}$ into a *-algebra.

Moreover, one can show that for any graph ${\displaystyle E}$, the Leavitt path algebra ${\displaystyle L_{\mathbb {C} }(E)}$ is isomorphic to a dense *-subalgebra of the graph C*-algebra ${\displaystyle C^{*}(E)}$.

Leavitt path algebras has been computed for many graphs, and the following table shows some particular graphs and their Leavitt path algebras. We use the convention that a double arrow drawn from one vertex to another and labeled ${\displaystyle \infty }$ indicates that there are a countably infinite number of edges from the first vertex to the second.

Directed graph ${\displaystyle E}$	Leavitt path algebra ${\displaystyle L_{K}(E)}$
${\displaystyle \bullet }$	${\displaystyle K}$, the underlying field
	${\displaystyle K[x,x^{-1}]}$, the Laurent polynomials with coefficients in ${\displaystyle K}$
${\displaystyle v_{1}\longrightarrow v_{2}\longrightarrow \cdots \longrightarrow v_{n-1}\longrightarrow v_{n}}$	${\displaystyle M_{n}(K)}$, the ${\displaystyle n\times n}$ matrices with entries in ${\displaystyle K}$
${\displaystyle \bullet \longrightarrow \bullet \longrightarrow \bullet \longrightarrow \cdots }$	${\displaystyle M_{\infty }{K}}$, the countably indexed, finitely supported matrices with entries in ${\displaystyle K}$
	${\displaystyle M_{n}(K[x,x^{-1}])}$, the ${\displaystyle n\times n}$ matrices with entries in ${\displaystyle K[x,x^{-1}]}$
	the Leavitt algebra ${\displaystyle L_{K}(n)}$
	${\displaystyle M_{\infty }{K}^{1}}$, the unitization of the algebra ${\displaystyle M_{\infty }{K}}$

As with graph C*-algebras, graph-theoretic properties of ${\displaystyle E}$ correspond to algebraic properties of ${\displaystyle L_{K}(E)}$. Interestingly, it is often the case that the graph properties of ${\displaystyle E}$ that are equivalent to an algebraic property of ${\displaystyle L_{K}(E)}$ are the same graph properties of ${\displaystyle E}$ that are equivalent to corresponding C*-algebraic property of ${\displaystyle C^{*}(E)}$, and moreover, many of the properties for ${\displaystyle L_{K}(E)}$ are independent of the field ${\displaystyle K}$.

The following table provides a short list of some of the more well-known equivalences. The reader may wish to compare this table with the corresponding table for graph C*-algebras.

Property of ${\displaystyle E}$	Property of ${\displaystyle L_{K}(E)}$
${\displaystyle E}$ is a finite, acylic graph.	${\displaystyle L_{K}(E)}$ is finite dimensional.
The vertex set ${\displaystyle E^{0}}$ is finite.	${\displaystyle L_{K}(E)}$ is unital (i.e., ${\displaystyle L_{K}(E)}$ contains a multiplicative identity).
${\displaystyle E}$ has no cycles.	${\displaystyle L_{K}(E)}$ is an ultramatrical ${\displaystyle K}$-algebra (i.e., a direct limit of finite-dimensional ${\displaystyle K}$-algebras).
${\displaystyle E}$ satisfies the following three properties: Condition (L), for each vertex ${\displaystyle v}$ and each infinite path ${\displaystyle \alpha }$ there exists a directed path from ${\displaystyle v}$ to a vertex on ${\displaystyle \alpha }$, and for each vertex ${\displaystyle v}$ and each singular vertex ${\displaystyle w}$ there exists a directed path from ${\displaystyle v}$ to ${\displaystyle w}$	${\displaystyle L_{K}(E)}$ is simple.
${\displaystyle E}$ satisfies the following three properties: Condition (L), for each vertex ${\displaystyle v}$ in ${\displaystyle E}$ there is a path from ${\displaystyle v}$ to a cycle.	Every left ideal of ${\displaystyle L_{K}(E)}$ contains an infinite idempotent. (When ${\displaystyle L_{K}(E)}$ is simple this is equivalent to ${\displaystyle L_{K}(E)}$ being a purely infinite ring.)

For a path ${\displaystyle \alpha :=e_{1}\ldots e_{n}}$ we let ${\displaystyle |\alpha |:=n}$ denote the length of ${\displaystyle \alpha }$. For each integer ${\displaystyle n\in \mathbb {Z} }$ we define ${\displaystyle L_{K}(E)_{n}:=\operatorname {span} _{K}\{s_{\alpha }s_{\beta }^{*}:|\alpha |-|\beta |=n\}}$. One can show that this defines a ${\displaystyle \mathbb {Z} }$-grading on the Leavitt path algebra ${\displaystyle L_{K}(E)}$ and that ${\displaystyle L_{K}(E)=\bigoplus _{n\in \mathbb {Z} }L_{K}(E)_{n}}$ with ${\displaystyle L_{K}(E)_{n}}$ being the component of homogeneous elements of degree ${\displaystyle n}$. It is important to note that the grading depends on the choice of the generating Cuntz-Krieger ${\displaystyle E}$-family ${\displaystyle \{s_{e},s_{e}^{*},p_{v}:e\in E^{1},v\in E^{0}\}}$. The grading on the Leavitt path algebra ${\displaystyle L_{K}(E)}$ is the algebraic analogue of the gauge action on the graph C*-algebra ${\displaystyle C*(E)}$, and it is a fundamental tool in analyzing the structure of ${\displaystyle L_{K}(E)}$.

There are two well-known uniqueness theorems for Leavitt path algebras: the graded uniqueness theorem and the Cuntz-Krieger uniqueness theorem. These are analogous, respectively, to the gauge-invariant uniqueness theorem and Cuntz-Krieger uniqueness theorem for graph C*-algebras. Formal statements of the uniqueness theorems are as follows:

The Graded Uniqueness Theorem: Fix a field ${\displaystyle K}$. Let ${\displaystyle E}$ be a graph, and let ${\displaystyle L_{K}(E)}$ be the associated Leavitt path algebra. If ${\displaystyle A}$ is a graded ${\displaystyle K}$-algebra and ${\displaystyle \phi :L_{K}(E)\to A}$ is a graded algebra homomorphism with ${\displaystyle \phi (p_{v})\neq 0}$ for all ${\displaystyle v\in E^{0}}$, then ${\displaystyle \phi }$ is injective.

The Cuntz-Krieger Uniqueness Theorem: Fix a field ${\displaystyle K}$. Let ${\displaystyle E}$ be a graph satisfying Condition (L), and let ${\displaystyle L_{K}(E)}$ be the associated Leavitt path algebra. If ${\displaystyle A}$ is a ${\displaystyle K}$-algebra and ${\displaystyle \phi :L_{K}(E)\to A}$ is an algebra homomorphism with ${\displaystyle \phi (p_{v})\neq 0}$ for all ${\displaystyle v\in E^{0}}$, then ${\displaystyle \phi }$ is injective.

We use the term ideal to mean "two-sided ideal" in our Leavitt path algebras. The ideal structure of ${\displaystyle L_{K}(E)}$ can be determined from ${\displaystyle E}$. A subset of vertices ${\displaystyle H\subseteq E^{0}}$ is called hereditary if for all ${\displaystyle e\in E^{1}}$, ${\displaystyle s(e)\in H}$ implies ${\displaystyle r(e)\in H}$. A hereditary subset ${\displaystyle H}$ is called saturated if whenever ${\displaystyle v}$ is a regular vertex with ${\displaystyle r(s^{-1}(v))\subseteq H}$, then ${\displaystyle v\in H}$. The saturated hereditary subsets of ${\displaystyle E}$ are partially ordered by inclusion, and they form a lattice with meet ${\displaystyle H_{1}\wedge H_{2}:=H_{1}\cap H_{2}}$ and join ${\displaystyle H_{1}\vee H_{2}}$ defined to be the smallest saturated hereditary subset containing ${\displaystyle H_{1}\cup H_{2}}$.

If ${\displaystyle H}$ is a saturated hereditary subset, ${\displaystyle I_{H}}$ is defined to be two-sided ideal in ${\displaystyle L_{K}(E)}$ generated by ${\displaystyle \{p_{v}:v\in H\}}$. A two-sided ideal ${\displaystyle I}$ of ${\displaystyle L_{K}(E)}$ is called a graded ideal if the ${\displaystyle I}$ has a ${\displaystyle \mathbb {Z} }$-grading ${\displaystyle I=\bigoplus _{n\in \mathbb {Z} }I_{n}}$ and ${\displaystyle I_{n}=L_{K}(E)_{n}\cap I}$ for all ${\displaystyle n\in \mathbb {Z} }$. The graded ideals are partially ordered by inclusion and form a lattice with meet ${\displaystyle I_{1}\wedge I_{2}:=I_{1}\cap I_{2}}$ and joint ${\displaystyle I_{1}\vee I_{2}}$ defined to be the ideal generated by ${\displaystyle I_{1}\cup I_{2}}$. For any saturated hereditary subset ${\displaystyle H}$, the ideal ${\displaystyle I_{H}}$ is graded.

The following theorem describes how graded ideals of ${\displaystyle L_{K}(E)}$ correspond to saturated hereditary subsets of ${\displaystyle E}$.

Theorem: Fix a field ${\displaystyle K}$, and let ${\displaystyle E}$ be a row-finite graph. Then the following hold:

1. The function ${\displaystyle H\mapsto I_{H}}$ is a lattice isomorphism from the lattice of saturated hereditary subsets of ${\displaystyle E}$ onto the lattice of graded ideals of ${\displaystyle L_{K}(E)}$ with inverse given by ${\displaystyle I\mapsto \{v\in E^{0}:p_{v}\in I\}}$.
2. For any saturated hereditary subset ${\displaystyle H}$, the quotient ${\displaystyle L_{K}(E)/I_{H}}$ is ${\displaystyle *}$-isomorphic to ${\displaystyle L_{K}(E\setminus H)}$, where ${\displaystyle E\setminus H}$ is the subgraph of ${\displaystyle E}$ with vertex set ${\displaystyle (E\setminus H)^{0}:=E^{0}\setminus H}$ and edge set ${\displaystyle (E\setminus H)^{1}:=E^{1}\setminus r^{-1}(H)}$.
3. For any saturated hereditary subset ${\displaystyle H}$, the ideal ${\displaystyle I_{H}}$ is Morita equivalent to ${\displaystyle L_{K}(E_{H})}$, where ${\displaystyle E_{H}}$ is the subgraph of ${\displaystyle E}$ with vertex set ${\displaystyle E_{H}^{0}:=H}$ and edge set ${\displaystyle E_{H}^{1}:=s^{-1}(H)}$.
4. If ${\displaystyle E}$ satisfies Condition (K), then every ideal of ${\displaystyle L_{K}(E)}$ is graded, and the ideals of ${\displaystyle L_{K}(E)}$ are in one-to-one correspondence with the saturated hereditary subsets of ${\displaystyle E}$.