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You also have the rather new field of Leavitt Path Algebras (in which I happen to be working right now), where you take a field $K$ and a directed graph $E$, generate its extended graph $E'$ (add to $E$ its own edges reversed, denoted as $e^*$ for every edge $e$), and compute the Leavitt path algebra of $E$, $L(E)$, as the path algebra $KE'$ modulo some relations called the Cuntz-Krieger relations, inherited from the $C^*$-algebras setting, concretely:

(CK1) $e^* f=\delta_{ef}$ for any two edges $e,f$ of $E'$.

(CK2) $\sum_{e\in s^{-1}(v)}ee^* = v$, for $v$ a vertex which emits a nonzero finite number of edges, and $s^{-1}(v)$ the set of those edges.

(One can look at (CK1) and (CK2) as an abstract generalization of the product of matrix units).

These associative algebras provide us simultaneously with a purely algebraic analog of $C*$-algebras C^*\$-algebras of graph and a generalization of the Leavitt algebras (some associative algebras which do not satisfy the IBN property).

The full matrix rings over $K$ of order $n$ then arise as the Leavitt path algebras of the graphs with $n$ (consecutive) vertices and $n-1$ arrows, one between every pair of consecutive vertices.

Another simple example of Leavitt path algebra is the ring of Laurent polynomials over $K$, $K[x,x^{-1}]$, which appears associated to the graph with one vertex and a single loop.

The theory of LPAs is a useful, and even beautifulone , becauseit allows :

• They provide simple, visually attractive representations of well-known algebras.

• They allow us to identify ring-theoretic look at their algebraic properties by means of associative algebras from the graph-theoretic combinatorial properties of their associated graphsin . This happens to equip us with some rather powerful tools.

• Conversely, they also enable "algebraic engineering", since they give us a visual and straightforward, visual way to construct new algebras, customized with any algebraic or ring-theoretic properties we may desire. For example, we can show an algebra generated by five elements such that it is exchange but not purely innitely simple, by constructing a particular (small) graph with some (easy) graph-theoretic features.

Some references:

• G. Abrams, G. Aranda Pino. "The Leavitt path algebra of a graph", J. Algebra 293 (2), 319-334 (2005). (Available at http://agt.cie.uma.es/~gonzalo/papers/AA1_Web.pdf).

• P. Ara, M.A. Moreno, E. Pardo. "Nonstable K-Theory for graph algebras", Algebra Repr. Th. DOI 10.1007/s10468-006-9044-z (electronic). (Available at http://www.springerlink.com/content/pu701474q5300m63/).

• G. Abrams, G. Aranda Pino, F. Perera, M. Siles Molina. "Chain conditions for Leavitt path algebras". (Available at http://agt.cie.uma.es/~gonzalo/papers/AAPS1_Web.pdf).

• K.R. Goodearl. "Leavitt path algebras and direct limits", Contemp. Math. 480 (2009), 165-187.

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You also have the rather new field of Leavitt Path Algebras (in which I happen to be working right now), where you take a field $K$ and a directed graph $E$, generate its extended graph $E'$ (add to $E$ its own edges reversed, denoted as $e^*$ for every edge $e$), and compute the Leavitt path algebra of $E$, $L(E)$, as the path algebra $KE'$ modulo some relations called the Cuntz-Krieger relations, inherited from the $C^*$-algebras setting, concretely:

(CK1) $e^* f=\delta_{ef}$ for any two edges $e,f$ of $E'$.

(CK2) $\sum_{e\in s^{-1}(v)}ee^* = v$, for $v$ a vertex which emits a nonzero finite number of edges, and $s^{-1}(v)$ the set of those edges.

(One can look at (CK1) and (CK2) as an abstract generalization of the product of matrix units).

These associative algebras provide us simultaneously with a purely algebraic analog of $C*$-algebras of graph and a generalization of the Leavitt algebras (some associative algebras which do not satisfy the IBN property).

The full matrix rings over $K$ of order $n$ then arise as the Leavitt path algebras of the graphs with $n$ (consecutive) vertices and $n-1$ arrows, one between every pair of consecutive vertices.

Another simple example of Leavitt path algebra is the ring of Laurent polynomials over $K$, $K[x,x^{-1}]$, which appears associated to the graph with one vertex and a single loop.

The theory of LPAs is a beautiful one because it allows us to identify ring-theoretic properties of associative algebras from the graph-theoretic properties of their associated graphs in a visual and straightforward way.

Some references:

G. Abrams, G. Aranda Pino. "The Leavitt path algebra of a graph", J. Algebra 293 (2), 319-334 (2005). (Available at http://agt.cie.uma.es/~gonzalo/papers/AA1_Web.pdf).

P. Ara, M.A. Moreno, E. Pardo. "Nonstable K-Theory for graph algebras", Algebra Repr. Th. DOI 10.1007/s10468-006-9044-z (electronic). (Available at http://www.springerlink.com/content/pu701474q5300m63/).

G. Abrams, G. Aranda Pino, F. Perera, M. Siles Molina. "Chain conditions for Leavitt path algebras". (Available at http://agt.cie.uma.es/~gonzalo/papers/AAPS1_Web.pdf).

K.R. Goodearl. "Leavitt path algebras and direct limits", Contemp. Math. 480 (2009), 165-187.
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