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.
consisting of a countable set of vertices
identifying the range and source of each edge, respectively.
A path is a finite sequence of edges
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.
Using the Cuntz–Krieger relations, one can show that Thus a typical element of
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.
Interestingly, it is often the case that the graph properties of
The following table provides a short list of some of the more well-known equivalences.
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
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
Formal statements of the uniqueness theorems are as follows: The Graded Uniqueness Theorem: Fix a field
be the associated Leavitt path algebra.
The Cuntz-Krieger Uniqueness Theorem: Fix a field
be the associated Leavitt path algebra.
are partially ordered by inclusion, and they form a lattice with meet
defined to be the smallest saturated hereditary subset containing
The graded ideals are partially ordered by inclusion and form a lattice with meet
The following theorem describes how graded ideals of
correspond to saturated hereditary subsets of