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.

History

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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]

Graph terminology

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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

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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

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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  
   , the underlying field
   , the Laurent polynomials with coefficients in  
   , the   matrices with entries in  
   , the countably indexed, finitely supported matrices with entries in  
   , the   matrices with entries in  
  the Leavitt algebra  
   , the unitization of the algebra  

Correspondence between graph and algebraic properties

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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

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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

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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

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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  .

References

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  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, Springer, London, 2017. Online Copy (PDF)
  4. ^ 2020 Mathematics Subject Classification (PDF)
  5. ^ Gene Abrams, Pere Ara, Mercedes Siles Molina (2017), Leavitt Path Algebras, Lecture Notes in Mathematics, vol. 2191, Springer, London, doi:10.1007/978-1-4471-7344-1, ISBN 978-1-4471-7343-4{{citation}}: CS1 maint: multiple names: authors list (link)