Matrices into path algebras

Question

I was thinking about quivers recently, and the following idea came to me.

Let e_i,j denote the matrix unit in M_n for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, …, n}: one directed edge E_i,j for each ordered pair (i, j), including self-loops E_i,i.

M_n(k) is then the quotient of the path algebra PΓ by a (rather large) ideal generated by "2-faces" of the simplex: E_i,jE_k,l = δ_j,kE_i,l.

In this language, for example, the Borel of upper triangular matrices corresponds to the ordered simplex inside Γ.

• Is this correspondence interesting?
• Can we transport Lie theoretic ideas about gl_n(k) to the quiver language? Should we?
• What happens if we quotient by a smaller ideal? Say, only reduce paths of length at least 3 (E_i,jE_k,lE_p,q = δ_j,kδ_l,pE_i,q).

My apologies in advance for these questions being vague.

Answer 1

There's a slightly different equivalence that is also useful. Consider the quiver with n elements, and an arrow E_i from i to i+1 and another F_i from i+1 to i for all i. The relations are then that E_i F_i = e_i and F_i E_i = e_{i+1}, where e_j is the j-th simple idempotent. This gets the same path algebra with fewer arrows and relations, but it has even less symmetry than your presentation.

A first answer to your question is that this perspective can often be useful. The reason I say this is because this perspective allows you to realize a lot of other quivers as subalgebras of matrices, and vice versa (for instance, the Borel subalgebra as the path algebra of a subquiver). It's not an extremely useful proving technique, but it can be a good way to produce a lot of quivers, especially when first learning about them.

Is it interesting? That's another question entirely. It's unfortunate that it picks out a basis in a necessary way, and so the GL_n action on M_n doesn't seem natural. I think the fact the the presentation I mention above is close to what is called a 'double quiver' is somewhat interesting. Especially if you like to think of a semisimple Lie algebra as something like the tangent bundle to the space of Borel subalgebras. Precisely, I mean that BB localization relates certain modules of g to D-modules on the space of Borel subalgebras, and so it is interesting to think of M_n as a deformation of the tangent bundle to U_n, the upper triangular matrices.

Answer 2

Somewhat related to this, you have the rather new field of Leavitt Path Algebras, which takes a field $K$ and a directed graph $E$, generates its extended graph $E'$ (add to $E$ its own edges reversed), and finally computes the Leavitt path algebra of $E$, $L(E)$, as the path algebra $KE'$ modulo some relations called (CK1) and (CK2), inherited from the $C*$-algebras setting.

These associative algebras provide us simultaneously with a purely algebraic analog of $C*$-algebras of graph and a generalization of the Leavitt algebras (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.