Consider the variety Comm_n of pairs of commuting n by n matrices, it is invariant under the action of GL_n on the space of pairs of all matrices, by simultaneous conjunction. The natural algebraic structure to associate with is equivariant

Quiver representation theory is a fundamental tool in modern representation theory of associative algebras. A cornerstone of this field is Gabriel’s theorem, which characterizes quivers with finitely many indecomposable representations as precisely those arising from Dynkin diagrams of type ADE.