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

Let e<sub>i,j</sub> denote the matrix unit in M<sub>n</sub> for 1 &le; i,j &le; n.  Let &Gamma; denote the complete quiver on vertices {1, &hellip;, n}:  one directed edge E<sub>i,j</sub> for each ordered pair (i, j), including self-loops E<sub>i,i</sub>.

M<sub>n</sub>(k) is then the quotient of the path algebra P&Gamma; by a (rather large) ideal generated by "2-faces" of the simplex":  E<sub>i,j</sub>E<sub>k,l</sub> = &delta;<sub>j,k</sub>E<sub>i,l</sub>.

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

&bull;  Is this correspondence **_interesting_**?

&bull;  Can we transport Lie theoretic ideas about gl<sub>n</sub>(k) to the quiver language?  Should we?

&bull;  What happens if we quotient by a smaller ideal?  Say, only reduce paths of length at least 3 (E<sub>i,j</sub>E<sub>k,l</sub>E<sub>p,q</sub> = &delta;<sub>j,k</sub>&delta;<sub>l,p</sub>E<sub>i,q</sub>).

My apologies in advance for these question being vague.