On the Orthogonal Projections

For any E-rigid presentation e, we construct an orthogonal projection functor to rep (e^) left adjoint to the natural embedding. We establish a bijection between presentations in rep (e^) and presentations compatible with e. For quivers with potentials, we show that rep (e^) forms a module category of another quiver with potential. We derive mutation formulas for the δ-vectors of positive and negative complements and the dimension vectors of simple modules in rep (e^), enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.