12
$\begingroup$

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

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

Mn(k) is then the quotient of the path algebra PΓ by a (rather large) ideal generated by "2-faces" of the simplex: Ei,jEk,l = δj,kEi,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 gln(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 (Ei,jEk,lEp,q = δj,kδl,pEi,q).

My apologies in advance for these questions being vague.

$\endgroup$
1
  • $\begingroup$ Removed grossly unnecessary "soft-question" tag! If it's actually maths, no need to be overly humble. $\endgroup$ Oct 27, 2009 at 1:33

2 Answers 2

4
$\begingroup$

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.

$\endgroup$
0
1
$\begingroup$

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.

$\endgroup$
1

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.