Leavitt path algebra

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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 [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 for this topic is the book by Gene Abrams, Pere Ara and Mercedes Siles Molina, Leavitt path algebras [5] .

Contents

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 consisting of a countable set of vertices , a countable set of edges , and maps identifying the range and source of each edge, respectively. A vertex is called a sink when ; i.e., there are no edges in with source . A vertex is called an infinite emitter when is infinite; i.e., there are infinitely many edges in with source . 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 is regular if and only if the number of edges in with source 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 with for all . An infinite path is a countably infinite sequence of edges with for all . A cycle is a path with , and an exit for a cycle is an edge such that and for some . A cycle is called a simple cycle if for all .

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 . A Cuntz–Krieger -family is a collection in a -algebra such that the following three relations (called the Cuntz–Krieger relations) are satisfied:

(CK0) for all ,
(CK1) for all ,
(CK2) whenever is a regular vertex, and
(CK3) for all .

The Leavitt path algebra corresponding to , denoted by , is defined to be the -algebra generated by a Cuntz–Krieger -family that is universal in the sense that whenever is a Cuntz–Krieger -family in a -algebra there exists a -algebra homomorphism with for all , for all , and for all .

We define for , and for a path we define and . Using the Cuntz–Krieger relations, one can show that

Thus a typical element of has the form for scalars and paths in . If is a field with an involution (e.g., when ), then one can define a *-operation on by that makes into a *-algebra.

Moreover, one can show that for any graph , the Leavitt path algebra is isomorphic to a dense *-subalgebra of the graph C*-algebra .

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


Directed graph Leavitt path algebra
Graph-single-vertex.jpg , the underlying field
Graph-one-edge-one-vertex.jpg , the Laurent polynomials with coefficients in
Line-graph.jpg , the matrices with entries in
Compacts-graph.jpg , the countably indexed, finitely supported matrices with entries in
C-M-n-graph.jpg , the matrices with entries in
O-n-graph.jpg the Leavitt algebra
K-unitization-graph.jpg , the unitization of the algebra

Correspondence between graph and algebraic properties

As with graph C*-algebras, graph-theoretic properties of correspond to algebraic properties of . Interestingly, it is often the case that the graph properties of that are equivalent to an algebraic property of are the same graph properties of that are equivalent to corresponding C*-algebraic property of , and moreover, many of the properties for are independent of the field .

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

The grading

For a path we let denote the length of . For each integer we define . One can show that this defines a -grading on the Leavitt path algebra and that with being the component of homogeneous elements of degree . It is important to note that the grading depends on the choice of the generating Cuntz-Krieger -family . The grading on the Leavitt path algebra is the algebraic analogue of the gauge action on the graph C*-algebra , and it is a fundamental tool in analyzing the structure of .

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 . Let be a graph, and let be the associated Leavitt path algebra. If is a graded -algebra and is a graded algebra homomorphism with for all , then is injective.

The Cuntz-Krieger Uniqueness Theorem: Fix a field . Let be a graph satisfying Condition (L), and let be the associated Leavitt path algebra. If is a -algebra and is an algebra homomorphism with for all , then is injective.

Ideal structure

We use the term ideal to mean "two-sided ideal" in our Leavitt path algebras. The ideal structure of can be determined from . A subset of vertices is called hereditary if for all , implies . A hereditary subset is called saturated if whenever is a regular vertex with , then . The saturated hereditary subsets of are partially ordered by inclusion, and they form a lattice with meet and join defined to be the smallest saturated hereditary subset containing .

If is a saturated hereditary subset, is defined to be two-sided ideal in generated by . A two-sided ideal of is called a graded ideal if the has a -grading and for all . The graded ideals are partially ordered by inclusion and form a lattice with meet and joint defined to be the ideal generated by . For any saturated hereditary subset , the ideal is graded.

The following theorem describes how graded ideals of correspond to saturated hereditary subsets of .

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

  1. The function is a lattice isomorphism from the lattice of saturated hereditary subsets of onto the lattice of graded ideals of with inverse given by .
  2. For any saturated hereditary subset , the quotient is -isomorphic to , where is the subgraph of with vertex set and edge set .
  3. For any saturated hereditary subset , the ideal is Morita equivalent to , where is the subgraph of with vertex set and edge set .
  4. If satisfies Condition (K), then every ideal of is graded, and the ideals of are in one-to-one correspondence with the saturated hereditary subsets of .

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References

  1. Abrams, Gene; Aranda Pino, Gonzalo; The Leavitt path algebra of a graph. J. Algebra 293 (2005), no. 2, 319–334.
  2. Pere Ara, María A. Moreno, and Enrique Pardo. Nonstable K-theory for graph algebras. Algebr. Represent. Theory, 10(2):157–178, 2007.
  3. Sec. 1.7 of Leavitt path algebras. Lecture Notes in Mathematics, 2191. Springer, London, 2017. xiii+287 pp. ISBN   978-1-4471-7343-4; 978-1-4471-7344-1. Online Copy (PDF)
  4. 2020 Mathematics Subject Classification (PDF)
  5. Gene Abrams, Pere Ara, Mercedes Siles Molina. Leavitt path algebras. Lecture Notes in Mathematics, 2191. Springer, London, 2017., Lecture Notes in Mathematics, vol. 2191, 2017, doi:10.1007/978-1-4471-7344-1, ISBN   978-1-4471-7343-4