Leavitt path algebra

In mathematics, a Leavitt path algebra is a universal algebra constructed from a directed graph. The Leavitt path algebras generalize the Leavitt algebras and may also be considered as algebraic analogues of the graph C*-algebras. Leavitt path algebras were simultaneously introduced in 2005 by Gene Abrams and Gonzalo Aranda Pino as well as by Pere Ara, María Moreno, and Enrique Pardo, with neither of the two groups aware of the other's work. 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.
Graph terminology.
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 formula_1 consisting of a countable set of vertices formula_2, a countable set of edges formula_3, and maps formula_4 identifying the range and source of each edge, respectively. A vertex formula_5 is called a sink when formula_6; i.e., there are no edges in formula_7 with source formula_8. A vertex formula_5 is called an infinite emitter when formula_10 is infinite; i.e., there are infinitely many edges in formula_7 with source formula_8. 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 formula_8 is regular if and only if the number of edges in formula_7 with source formula_8 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 formula_16 with formula_17 for all formula_18. An infinite path is a countably infinite sequence of edges formula_19 with formula_17 for all formula_21. A cycle is a path formula_16 with formula_23, and an exit for a cycle formula_16 is an edge formula_25 such that formula_26 and formula_27 for some formula_28. A cycle formula_16 is called a simple cycle if formula_30 for all formula_31.
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.
The Cuntz–Krieger relations and the universal property.
Fix a field formula_32. A Cuntz–Krieger formula_7-family is a collection formula_34 in a formula_32-algebra such that the following three relations (called the Cuntz–Krieger relations) are satisfied:
(CK0) formula_36 for all formula_37,
(CK1) formula_38 for all formula_39,
(CK2) formula_40 whenever formula_8 is a regular vertex, and
(CK3) formula_42 for all formula_43.
The Leavitt path algebra corresponding to formula_7, denoted by formula_45, is defined to be the formula_32-algebra generated by a Cuntz–Krieger formula_7-family that is universal in the sense that whenever formula_48 is a Cuntz–Krieger formula_7-family in a formula_32-algebra formula_51 there exists a formula_32-algebra homomorphism formula_53 with formula_54 for all formula_43, formula_56 for all formula_43, and formula_58 for all formula_5. 
We define formula_60 for formula_5, and for a path formula_62 we define formula_63 and formula_64. Using the Cuntz–Krieger relations, one can show that 
formula_65 
Thus a typical element of formula_45 has the form formula_67 for scalars formula_68 and paths formula_69 in formula_7. If formula_32 is a field with an involution formula_72 (e.g., when formula_73), then one can define a *-operation on formula_45 by formula_75 that makes formula_45 into a *-algebra.
Moreover, one can show that for any graph formula_7, the Leavitt path algebra formula_78 is isomorphic to a dense *-subalgebra of the graph C*-algebra formula_79.
Examples.
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 formula_80 indicates that there are a countably infinite number of edges from the first vertex to the second.
Correspondence between graph and algebraic properties.
As with graph C*-algebras, graph-theoretic properties of formula_7 correspond to algebraic properties of formula_45. Interestingly, it is often the case that the graph properties of formula_7 that are equivalent to an algebraic property of formula_45 are the same graph properties of formula_7 that are equivalent to corresponding C*-algebraic property of formula_79, and moreover, many of the properties for formula_45 are independent of the field formula_32.
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.
The grading.
For a path formula_62 we let formula_90 denote the length of formula_91. For each integer formula_92 we define formula_93. One can show that this defines a formula_94-grading on the Leavitt path algebra formula_45 and that formula_96 with formula_97 being the component of homogeneous elements of degree formula_98. It is important to note that the grading depends on the choice of the generating Cuntz-Krieger formula_7-family formula_100. The grading on the Leavitt path algebra formula_45 is the algebraic analogue of the gauge action on the graph C*-algebra formula_102, and it is a fundamental tool in analyzing the structure of formula_45.
The uniqueness theorems.
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 formula_32. Let formula_7 be a graph, and let formula_45 be the associated Leavitt path algebra. If formula_51 is a graded formula_32-algebra and formula_53 is a graded algebra homomorphism with formula_110 for all formula_5, then formula_112 is injective.
The Cuntz-Krieger Uniqueness Theorem: Fix a field formula_32. Let formula_7 be a graph satisfying Condition (L), and let formula_45 be the associated Leavitt path algebra. If formula_51 is a formula_32-algebra and formula_53 is an algebra homomorphism with formula_110 for all formula_5, then formula_112 is injective.
Ideal structure.
We use the term ideal to mean "two-sided ideal" in our Leavitt path algebras. The ideal structure of formula_45 can be determined from formula_7. A subset of vertices formula_124 is called hereditary if for all formula_43, formula_126 implies formula_127. A hereditary subset formula_128 is called saturated if whenever formula_8 is a regular vertex with formula_130, then formula_131. The saturated hereditary subsets of formula_7 are partially ordered by inclusion, and they form a lattice with meet formula_133 and join formula_134 defined to be the smallest saturated hereditary subset containing formula_135.
If formula_128 is a saturated hereditary subset, formula_137 is defined to be two-sided ideal in formula_45 generated by formula_139. A two-sided ideal formula_140 of formula_45 is called a graded ideal if the formula_140 has a formula_94-grading formula_144 and formula_145 for all formula_92. The graded ideals are partially ordered by inclusion and form a lattice with meet formula_147 and joint formula_148 defined to be the ideal generated by formula_149. For any saturated hereditary subset formula_128, the ideal formula_137 is graded.
The following theorem describes how graded ideals of formula_45 correspond to saturated hereditary subsets of formula_7.
Theorem: Fix a field formula_32, and let formula_7 be a row-finite graph. Then the following hold: