Non-commutative Rank and Semi-stability of Quiver Representations

Fortin and Reutenauer defined the non-commutative rank for a matrix with entries that are linear functions. The non-commutative rank is related to stability in invariant theory, non-commutative arithmetic circuits, and Edmonds' problem. We will generalize the non-commutative rank to the representation theory of quivers and define non-commutative Hom and Ext spaces. We will relate these new notions to King's criterion for $\sigma$-stability of quiver representations, and the general Hom and Ext spaces studied by Schofield. We discuss polynomial time algorithms that compute the non-commutative Homs and Exts and find an optimal witness for the $\sigma$-semi-stability of a quiver representation.