Leavitt path algebra

Leavitt path algebras were simultaneously introduced in 2005 by Gene Abrams and Gonzalo Aranda PinoAbrams, Gene; Aranda Pino, Gonzalo; The Leavitt path algebra

of a graph. J. Algebra 293 (2005), no. 2, 319–334. as well as by Pere Ara, María Moreno, and Enrique Pardo,Pere Ara, María A. Moreno, and Enrique Pardo. Nonstable K-theory for

graph algebras. Algebr. Represent. Theory 10(2):157–178, 2007. with neither of the two groups aware of the other's work.Sec. 1.7 of Leavitt Path Algebras, Springer, London, 2017. {{citation|url=https://academics.uccs.edu/gabrams/documents/versionsenttoSpringer4April2017.pdf|title=Online Copy}} 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.{{citation|url=https://mathscinet.ams.org/msnhtml/msc2020.pdf|title=2020 Mathematics Subject Classification}}

The basic reference is the book Leavitt Path Algebras. {{citation|url=https://link.springer.com/book/10.1007/978-1-4471-7344-1 |author=Gene Abrams, Pere Ara, Mercedes Siles Molina |title=Leavitt Path Algebras|publisher=Springer, London|series=Lecture Notes in Mathematics |year=2017 |volume=2191 |doi=10.1007/978-1-4471-7344-1 |isbn=978-1-4471-7343-4 |url-access=subscription }}

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 $E=(E^0,\; E^1,\; r,\; s)$ consisting of a countable set of vertices $E^0$, a countable set of edges $E^1$, and maps $r,\; s\; :\; E^1\; \backslash rightarrow\; E^0$ identifying the range and source of each edge, respectively. A vertex $v\; \backslash in\; E^0$ is called a sink when $s^\{-1\}(v)\; =\; \backslash emptyset$; i.e., there are no edges in $E$ with source $v$. A vertex $v\; \backslash in\; E^0$ is called an infinite emitter when $s^\{-1\}(v)$ is infinite; i.e., there are infinitely many edges in $E$ with source $v$. 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 $v$ is regular if and only if the number of edges in $E$ with source $v$ 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 $e\_1\; e\_2\; \backslash ldots\; e\_n$ with $r(e\_i)\; =\; s(e\_\{i+1\})$ for all $1\; \backslash leq\; i\; \backslash leq\; n-1$. An infinite path is a countably infinite sequence of edges $e\_1\; e\_2\; \backslash ldots$ with $r(e\_i)\; =\; s(e\_\{i+1\})$ for all $i\; \backslash in\; \backslash mathbb\{N\}$. A cycle is a path $e\_1\; e\_2\; \backslash ldots\; e\_n$ with $r(e\_n)\; =\; s(e\_1)$, and an exit for a cycle $e\_1\; e\_2\; \backslash ldots\; e\_n$ is an edge $f\; \backslash in\; E^1$ such that $s(f)\; =\; s(e\_i)$ and $f\; \backslash neq\; e\_i$ for some $1\; \backslash leq\; i\; \backslash leq\; n$. A cycle $e\_1\; e\_2\; \backslash ldots\; e\_n$ is called a simple cycle if $s(e\_i)\; \backslash neq\; s(e\_1)$ for all $2\; \backslash leq\; i\; \backslash leq\; n$.

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 $K$. A Cuntz–Krieger $E$-family is a collection $\backslash \{\; s\_e^*,\; s\_e,\; p\_v\; :\; e\; \backslash in\; E^1,\; v\; \backslash in\; E^0\; \backslash \}$ in a $K$-algebra such that the following three relations (called the Cuntz–Krieger relations) are satisfied:

: (CK0) for all $v,\; w\; \backslash in\; E^0$,

: (CK1) for all $e,\; f\; \backslash in\; E^1$,

: (CK2) $p\_v\; =\; \backslash sum\_\{s(e)=v\}\; s\_e\; s\_e^*$ whenever $v$ is a regular vertex, and

: (CK3) $p\_\{s(e)\}\; s\_e\; =\; s\_e$ for all $e\; \backslash in\; E^1$.

The Leavitt path algebra corresponding to $E$, denoted by $L\_K(E)$, is defined to be the $K$-algebra generated by a Cuntz–Krieger $E$-family that is universal in the sense that whenever $\backslash \{\; t\_e,\; t\_e^*,\; q\_v\; :\; e\; \backslash in\; E^1,\; v\; \backslash in\; E^0\; \backslash \}$ is a Cuntz–Krieger $E$-family in a $K$-algebra $A$ there exists a $K$-algebra homomorphism $\backslash phi\; :\; L\_K(E)\; \backslash to\; A$ with $\backslash phi(s\_e)\; =\; t\_e$ for all $e\; \backslash in\; E^1$, $\backslash phi(s\_e^*)\; =\; t\_e^*$ for all $e\; \backslash in\; E^1$, and $\backslash phi(p\_v)=q\_v$ for all $v\; \backslash in\; E^0$.

We define $p\_v^*\; :=\; p\_v$ for $v\; \backslash in\; E^0$, and for a path $\backslash alpha\; :=\; e\_1\; \backslash ldots\; e\_n$ we define $s\_\backslash alpha\; :=\; s\_\{e\_1\}\; \backslash ldots\; s\_\{e\_n\}$ and $s\_\backslash alpha^*\; :=\; s\_\{e\_n\}^*\; \backslash ldots\; s\_\{e\_1\}^*$. Using the Cuntz–Krieger relations, one can show that

:$L\_K(E)\; =\; \backslash operatorname\{span\}\_K\; \backslash \{\; s\_\backslash alpha\; s\_\backslash beta^*\; :\; \backslash alpha\; \backslash text\{\; and\; \}\; \backslash beta\; \backslash text\{\; are\; paths\; in\; \}\; E\; \backslash \}.$

Thus a typical element of $L\_K(E)$ has the form $\backslash sum\_\{i=1\}^n\; \backslash lambda\_i\; s\_\{\backslash alpha\_i\}\; s\_\{\backslash beta\_i\}^*$ for scalars $\backslash lambda\_1,\; \backslash ldots,\backslash lambda\_n\; \backslash in\; K$ and paths $\backslash alpha\_1,\; \backslash ldots,\; \backslash alpha\_n,\; \backslash beta\_1,\; \backslash ldots,\; \backslash beta\_n$ in $E$. If $K$ is a field with an involution $\backslash lambda\; \backslash mapsto\; \backslash overline\{\backslash lambda\}$ (e.g., when $K=\backslash mathbb\{C\}$), then one can define a *-operation on $L\_K(E)$ by $\backslash sum\_\{i=1\}^n\; \backslash lambda\_i\; s\_\{\backslash alpha\_i\}\; s\_\{\backslash beta\_i\}^*\; \backslash mapsto\; \backslash sum\_\{i=1\}^n\; \backslash overline\{\backslash lambda\_i\}\; s\_\{\backslash beta\_i\}\; s\_\{\backslash alpha\_i\}^*$ that makes $L\_K(E)$ into a *-algebra.

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

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 $\backslash infty$ indicates that there are a countably infinite number of edges from the first vertex to the second.

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

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.

For a path $\backslash alpha\; :=\; e\_1\; \backslash ldots\; e\_n$ we let $|\backslash alpha|\; :=\; n$ denote the length of $\backslash alpha$. For each integer $n\; \backslash in\; \backslash mathbb\{Z\}$ we define $L\_K(E)\_n\; :=\; \backslash operatorname\{span\}\_K\; \backslash \{\; s\_\backslash alpha\; s\_\backslash beta^*\; :\; |\backslash alpha|-|\backslash beta|\; =\; n\; \backslash \}$. One can show that this defines a $\backslash mathbb\{Z\}$-grading on the Leavitt path algebra $L\_K(E)$ and that $L\_K(E)\; =\; \backslash bigoplus\_\{n\; \backslash in\; \backslash mathbb\{Z\}\}\; L\_K(E)\_n$ with $L\_K(E)\_n$ being the component of homogeneous elements of degree $n$. It is important to note that the grading depends on the choice of the generating Cuntz-Krieger $E$-family $\backslash \{\; s\_e,\; s\_e^*,\; p\_v\; :\; e\; \backslash in\; E^1,\; v\; \backslash in\; E^0\; \backslash \}$. The grading on the Leavitt path algebra $L\_K(E)$ is the algebraic analogue of the gauge action on the graph C*-algebra $C*(E)$, and it is a fundamental tool in analyzing the structure of $L\_K(E)$.

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 $K$. Let $E$ be a graph, and let $L\_K(E)$ be the associated Leavitt path algebra. If $A$ is a graded $K$-algebra and $\backslash phi\; :\; L\_K(E)\; \backslash to\; A$ is a graded algebra homomorphism with $\backslash phi(p\_v)\; \backslash neq\; 0$ for all $v\; \backslash in\; E^0$, then $\backslash phi$ is injective.

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

We use the term ideal to mean "two-sided ideal" in our Leavitt path algebras. The ideal structure of $L\_K(E)$ can be determined from $E$. A subset of vertices $H\; \backslash subseteq\; E^0$ is called hereditary if for all $e\; \backslash in\; E^1$, $s(e)\; \backslash in\; H$ implies $r(e)\; \backslash in\; H$. A hereditary subset $H$ is called saturated if whenever $v$ is a regular vertex with $r(s^\{-1\}(v))\; \backslash subseteq\; H$, then $v\; \backslash in\; H$. The saturated hereditary subsets of $E$ are partially ordered by inclusion, and they form a lattice with meet $H\_1\; \backslash wedge\; H\_2\; :=\; H\_1\; \backslash cap\; H\_2$ and join $H\_1\; \backslash vee\; H\_2$ defined to be the smallest saturated hereditary subset containing $H\_1\; \backslash cup\; H\_2$.

If $H$ is a saturated hereditary subset, $I\_H$ is defined to be two-sided ideal in $L\_K(E)$ generated by $\backslash \{\; p\_v\; :\; v\; \backslash in\; H\; \backslash \}$. A two-sided ideal $I$ of $L\_K(E)$ is called a graded ideal if the $I$ has a $\backslash mathbb\{Z\}$-grading $I\; =\; \backslash bigoplus\_\{n\; \backslash in\; \backslash mathbb\{Z\}\}\; I\_n$ and $I\_n\; =\; L\_K(E)\_n\; \backslash cap\; I$ for all $n\; \backslash in\; \backslash mathbb\{Z\}$. The graded ideals are partially ordered by inclusion and form a lattice with meet $I\_1\; \backslash wedge\; I\_2\; :=\; I\_1\; \backslash cap\; I\_2$ and joint $I\_1\; \backslash vee\; I\_2$ defined to be the ideal generated by $I\_1\; \backslash cup\; I\_2$. For any saturated hereditary subset $H$, the ideal $I\_H$ is graded.

The following theorem describes how graded ideals of $L\_K(E)$ correspond to saturated hereditary subsets of $E$.

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

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

Category:Algebras

Category:Ring theory