Quivers with relations arising from Koszul algebras of $\mathfrak g$-invariants

Let $\mathfrak g$ be a complex simple Lie algebra and let $\Psi$ be an extremal set of positive roots. One associates with $\Psi$ an infinite dimensional Koszul algebra $\bold S_\Psi^{\lie g}$ which is a graded subalgebra of the locally finite part of $((\bold V)^{op}\tensor S(\lie g))^{\lie g}$, where $\bold V$ is the direct sum of all simple finite dimensional $\lie g$-modules. We describe the structure of the algebra $\bold S_\Psi^{\lie g}$ explicitly in terms of an infinite quiver with relations for $\lie g$ of types $A$ and $C$. We also describe several infinite families of quivers and finite dimensional algebras arising from this construction.