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.