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.



Matrices into path algebras
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.


