Quiver Demazure algebras I: Residue descent, finite quotients, and PBW bases

We study quiver Demazure algebras through residue covers and root-power quotients. The cover determined by the zero-root subsystem identifies the additive algebra with a finite fixed algebra of Sauter’s generalized quiver Hecke construction. Her basis theorem then gives additive PBW in every finite Weyl type, for arbitrary admissible residue data and uniform quivers. A sequence-cover argument gives global multiplicative PBW in simply-laced type. Under a directed-path condition, root-power quotients are finite dimensional and the formal exponential identifies their additive and multiplicative forms. For arrow-free quivers, we reduce mixed-residue quotients to matrix algebras over finite skew group rings, obtain sharp nonvanishing criteria from coinvariant geometry, and determine their simple modules. In a mixed type-$D_4$ family we compute an explicit coefficient presentation, dimensions, Loewy lengths, and Cartan matrices. We also specify the direct and finite-cover comparisons with Liu’s algebras.